Part 1. Infinite series and logarithmic integrals associated to   differentiation with respect to parameters of the Whittaker   $\mathrm{M}_{\kappa ,\mu }\left( x\right) $ function

Alexander Apelblat; Juan Luis Gonz\'alez-Santander

arXiv:2302.13776·math.CA·April 28, 2023

Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{\kappa ,\mu }\left( x\right) $ function

Alexander Apelblat, Juan Luis Gonz\'alez-Santander

PDF

Open Access

TL;DR

This paper derives formulas for the derivatives of the Whittaker function with respect to its parameters, expressing them as infinite sums and integrals, and applies these results to obtain reduction formulas and evaluate related integrals.

Contribution

It provides new closed-form expressions and reduction formulas for parameter derivatives of the Whittaker and confluent hypergeometric functions, including integral representations.

Findings

01

Derived parameter derivatives of the Whittaker function in closed-form.

02

Expressed derivatives as infinite sums involving digamma and gamma functions.

03

Obtained reduction formulas for Whittaker functions and related integrals.

Abstract

First derivatives of the Whittaker function $M_{κ, μ} (x)$ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $M_{κ, μ} (x)$ . These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed-form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function has…

Tables9

Table 1. Table 1: Derivative of M κ , μ subscript 𝑀 𝜅 𝜇 M_{\kappa,\mu} with respect to κ 𝜅 \kappa by using ( 19 ).

$κ$	$μ$	$\frac{\partial M_{κ, μ} (x)}{\partial κ}$
$- \frac{3}{4}$	$\frac{1}{4}$	$- \frac{2}{3} x^{7 / 4} e_{2}^{x / 2} F_{2} (1, 1; \frac{5}{2}, 2; - x)$
$- \frac{1}{2}$	$0$	$- \sqrt{x} e^{x / 2} [γ + \ln x + Shi (x) - Chi (x)]$
$- \frac{1}{4}$	$- \frac{1}{4}$	$- 2 x^{5 / 4} e_{2}^{x / 2} F_{2} (1, 1; \frac{3}{2}, 2; - x)$
$- \frac{1}{6}$	$- \frac{1}{3}$	$- 3 x^{7 / 6} e_{2}^{x / 2} F_{2} (1, 1; \frac{4}{3}, 2; - x)$
$0$	$\frac{1}{2}$	$e^{- x / 2} [Shi (x) + Chi (x) - \ln x - γ] - e^{x / 2} [Shi (x) - Chi (x) + \ln x + γ]$
$\frac{1}{6}$	$- \frac{2}{3}$	$3 x^{5 / 6} e_{2}^{x / 2} F_{2} (1, 1; \frac{2}{3}, 2; - x)$
$\frac{1}{2}$	$1$	$\begin{matrix} - \frac{2}{\sqrt{x}} {e^{x / 2} [γ + 1 + \ln x + Shi (x) - Chi (x)] \\ + e^{- x / 2} (x + 1) [γ - 1 + \ln x - Shi (x) - Chi (x)]} \end{matrix}$
$1$	$\frac{3}{2}$	$\begin{matrix} - \frac{3}{x} {e^{- x / 2} [(x^{2} + 2 x + 2) (\ln x - Shi (x) - Chi (x) + γ)] \\ + e^{x / 2} [2 \ln x + 2 Shi (x) - 2 Chi (x) + x + 2 γ + 3]} \end{matrix}$

Table 2. Table 2: Derivative of M κ , μ subscript 𝑀 𝜅 𝜇 M_{\kappa,\mu} with respect to κ 𝜅 \kappa by using ( 34 ).

$κ$	$μ$	$\frac{\partial M_{κ, μ} (x)}{\partial κ}$
$1$	$\frac{1}{2}$	$x e^{- x / 2} [\ln \| x \| - Ei (x) + γ - 1] + 2 \sinh (\frac{x}{2})$
$2$	$\frac{1}{2}$	$\frac{1}{2} x e^{- x / 2} {(2 - x) [\ln \| x \| - Ei (x) + γ - \frac{3}{2}] - e^{x} + 3} + \sinh (\frac{x}{2})$
$3$	$\frac{1}{2}$	$\begin{matrix} \frac{1}{6} x e^{- x / 2} [(x^{2} - 6 x + 6) (\ln \| x \| - Ei (x) + γ - \frac{11}{6}) \\ + (e^{x} - 5) (x - 2) - 3 e^{x} + 4] + \frac{2}{3} \sinh (\frac{x}{2}) \end{matrix}$

Table 3. Table 3: Derivative of M κ , μ subscript 𝑀 𝜅 𝜇 M_{\kappa,\mu} with respect to μ 𝜇 \mu by using ( 43 ).

$κ$	$μ$	$\frac{\partial M_{κ, μ} (x)}{\partial μ}$
$- \frac{3}{2}$	$1$	$\frac{1}{\sqrt{x}} {e^{x / 2} [x^{2} (Chi (x) - Shi (x) - γ) + \frac{3}{2} x^{2} - 2 x + 1] + e^{- x / 2} (x - 1)}$
$- 1$	$\frac{1}{2}$	$x e^{x / 2} [Chi (x) - Shi (x) - γ + 1] - 2 \sinh (\frac{x}{2})$
$- \frac{3}{4}$	$\frac{1}{4}$	$e^{x / 2} x^{3 / 4} [\ln x - \frac{2}{3} x_{2} F_{2} (\begin{matrix} 1, 1 \\ 2, \frac{5}{2} \end{matrix} \| - x)]$
$- \frac{1}{2}$	$0$	$e^{x / 2} \sqrt{x} [Chi (x) - Shi (x) - γ]$
$- \frac{1}{4}$	$- \frac{1}{4}$	$e^{x / 2} x^{1 / 4} [\ln x - 2 x_{2} F_{2} (\begin{matrix} 1, 1 \\ 2, \frac{3}{2} \end{matrix} \| - x)]$
$- \frac{1}{6}$	$- \frac{1}{3}$	$e^{x / 2} x^{1 / 6} [\ln x - 3 x_{2} F_{2} (\begin{matrix} 1, 1 \\ 2, \frac{4}{3} \end{matrix} \| - x)]$
$\frac{1}{6}$	$- \frac{2}{3}$	$e^{x / 2} x^{- 1 / 6} [\ln x + 3 x_{2} F_{2} (\begin{matrix} 1, 1 \\ 2, \frac{2}{3} \end{matrix} \| - x)]$

Table 4. Table 4: Derivative of M κ , μ subscript 𝑀 𝜅 𝜇 M_{\kappa,\mu} with respect to μ 𝜇 \mu by using ( 10 ).

$κ$	$μ$	$\frac{\partial M_{κ, μ} (x)}{\partial μ}$
$0$	$- \frac{1}{2}$	$[Chi (x) - γ] \cosh (\frac{x}{2}) - \frac{2}{x} \sinh^{3} (\frac{x}{2})$
$0$	$0$	$\sqrt{x} [(\ln 4 - γ) I_{0} (\frac{x}{2}) - K_{0} (\frac{x}{2})]$
$0$	$\frac{1}{4}$	$\begin{matrix} {\frac{x^{3 / 4}}{15}}_{0} F_{1} (; \frac{5}{4}; \frac{x^{2}}{16}) [x_{3}^{2} F_{4} (1, 1, \frac{3}{2}; \frac{7}{4}, 2, 2, \frac{9}{4}; \frac{x^{2}}{4}) + 15 (\ln x + 2)] \\ - \frac{2 π x}{Γ (\frac{1}{4})} I_{- \frac{1}{4}} {(\frac{x}{2})}_{2} F_{3} (\frac{1}{4}, \frac{3}{4}; \frac{5}{4}, \frac{5}{4}, \frac{3}{2}; \frac{x^{2}}{4}) \end{matrix}$
$0$	$\frac{1}{3}$	$\begin{matrix} \frac{x^{5 / 6}}{128} {{}_{0}F_{1} (; \frac{4}{3}; \frac{x^{2}}{16}) [9 x_{3}^{2} F_{4} (1, 1, \frac{3}{2}; \frac{5}{3}, 2, 2, \frac{7}{3}; \frac{x^{2}}{4}) + 64 (2 \ln x + 3)] \\ - 192_{0} F_{1} {(; \frac{2}{3}; \frac{x^{2}}{16})}_{2} F_{3} (\frac{1}{3}, \frac{5}{6}; \frac{4}{3}, \frac{4}{3}, \frac{5}{3}; \frac{x^{2}}{4})} \end{matrix}$
$0$	$\frac{1}{2}$	$2 [Chi (x) - γ + 2] \sinh (\frac{x}{2}) - 2 Shi (x) \cosh (\frac{x}{2})$
$0$	$\frac{2}{3}$	$\begin{matrix} \frac{x^{7 / 6}}{80} {{}_{0}F_{1} (; \frac{5}{3}; \frac{x^{2}}{16}) [9 x_{3}^{2} F_{4} (1, 1, \frac{3}{2}; \frac{4}{3}, 2, 2, \frac{8}{3}; \frac{x^{2}}{4}) + 80 \ln x + 60] \\ - 60_{0} F_{1} {(; \frac{1}{3}; \frac{x^{2}}{16})}_{2} F_{3} (\frac{2}{3}, \frac{7}{6}; \frac{5}{3}, \frac{5}{3}, \frac{7}{3}; \frac{x^{2}}{4})} \end{matrix}$
$0$	$\frac{3}{4}$	$\begin{matrix} {\frac{x^{5 / 4}}{21}}_{0} F_{1} (; \frac{7}{4}; \frac{x^{2}}{16}) [3 x_{3}^{2} F_{4} (1, 1, \frac{3}{2}; \frac{5}{4}, 2, 2, \frac{11}{4}; \frac{x^{2}}{4}) + 21 \ln x + 14] \\ - \frac{π x^{2}}{4 Γ (\frac{7}{4})} I_{- \frac{3}{4}} {(\frac{x}{2})}_{2} F_{3} (\frac{3}{4}, \frac{5}{4}; \frac{7}{4}, \frac{7}{4}, \frac{5}{2}; \frac{x^{2}}{4}) \end{matrix}$
$0$	$1$	$\begin{matrix} 4 \sqrt{x} {I_{1} (\frac{x}{2}) [1 - γ + \ln 4 - \frac{1}{2 \sqrt{π}} G_{1, 3}^{2, 1} (\frac{x^{2}}{4}; \frac{1}{2}; 0, 0, - 1)] \\ - K_{1} (\frac{x}{2}) [I_{0}^{2} (\frac{x}{2}) - I_{1}^{2} (\frac{x}{2}) - 1]} \end{matrix}$
$0$	$\frac{3}{2}$	$\begin{matrix} \frac{4}{x} {\sinh (\frac{x}{2}) [6 γ - 6 Chi (x) - 3 x Shi (x) - 28] \\ + \cosh (\frac{x}{2}) [(3 Chi (x) + 8 - 3 γ) x + 6 Shi (x)]} \end{matrix}$
$0$	$2$	$\begin{matrix} 32 \sqrt{x} {I_{2} (\frac{x}{2}) [\frac{3}{2} - γ + \ln 4 - \frac{1}{\sqrt{π}} G_{2, 4}^{3, 1} (\frac{x^{2}}{4}; \frac{1}{2}, 1; 0, 0, 2, - 2)] \\ + K_{2} (\frac{x}{2}) [2_{1} F_{2} (\frac{1}{2}; 1, 3; \frac{x^{2}}{4}) -_{2} F_{3} (\frac{1}{2}, 2; 1, 1, 3; \frac{x^{2}}{4}) - 1]} \end{matrix}$

Table 5. Table 5: Integral I 1 ( κ , μ ; x ) subscript 𝐼 1 𝜅 𝜇 𝑥 I_{1}(\kappa,\mu;x) for particular values of κ 𝜅 \kappa and μ 𝜇 \mu .

$κ$	$μ$	$I_{1} (κ, μ; x)$
$- \frac{1}{2}$	$1$	$\frac{1}{x^{2}} {e^{x} (1 - x) [\ln x + γ + Shi (x) - Chi (x)] + \ln x + γ - Chi (x) - Shi (x)}$
$- \frac{1}{2}$	$μ$	$- \frac{\sqrt{π}}{2} Γ (μ) {\frac{e^{x / 2} x^{1 / 2 - μ}}{μ} [I_{μ - 1 / 2} (\frac{x}{2}) + I_{μ + 1 / 2} (\frac{x}{2})] + \frac{2^{1 - 2 μ}}{Γ (μ + \frac{1}{2})} G^{(1)} (μ + 1; 2 μ + 1; x)}$
$\frac{1}{2}$	$1$	$\frac{1}{x^{2}} {(x + e^{x} + 1) [Chi (x) - \ln x - γ] + (x - e^{x} + 1) Shi (x)}$
$\frac{1}{2}$	$μ$	$\frac{\sqrt{π}}{2} Γ (μ) {\frac{e^{x / 2} x^{1 / 2 - μ}}{μ} [I_{μ - 1 / 2} (\frac{x}{2}) - I_{μ + 1 / 2} (\frac{x}{2})] - \frac{2^{1 - 2 μ}}{Γ (μ + \frac{1}{2})} G^{(1)} (μ; 2 μ + 1; x)}$
$1$	$μ$	$\begin{matrix} Γ (μ - \frac{1}{2}) {\frac{4 \sqrt{π} μ e^{x / 2} x^{- μ}}{4 μ^{2} - 1} [(2 μ - x + 1) I_{μ} (\frac{x}{2}) + x I_{μ + 1} (\frac{x}{2})] \\ - \frac{Γ (μ + \frac{2}{3})}{Γ (2 μ + 1)} G^{(1)} (μ - \frac{1}{2}; 2 μ + 1; x)} \end{matrix}$
$κ$	$0$	$π \sec (π κ) [π \tan (π κ) L_{κ - 1 / 2} (x) - G^{(1)} (\frac{1}{2} - κ; 1; x)]$
$κ$	$\frac{1}{2}$	$- π \csc (π κ) {{[π κ \cot (π κ) - 1]}_{1} F_{1} (1 - κ; 2; x) + κ G^{(1)} (1 - κ; 2; x)}$
$κ$	$κ$	$\sqrt{π} \frac{Γ (2 κ + \frac{1}{2})}{Γ (2 κ + 1)} {{[H_{2 κ - 1 / 2} + 2 \ln 2]}_{1} F_{1} (\frac{1}{2}; 2 κ + 1; x) - G^{(1)} (\frac{1}{2}; 2 κ + 1; x)}$
$\frac{1}{4}$	$\frac{1}{4}$	$\frac{4 e^{x} \ln 2}{\sqrt{x}} F (\sqrt{x}) - 2 G^{(1)} (\frac{1}{2}; \frac{3}{2}; x)$

Table 6. Table 6: Derivative of M κ , μ subscript 𝑀 𝜅 𝜇 M_{\kappa,\mu} with respect to κ 𝜅 \kappa by using ( 17 ).

$κ$	$μ$	$\frac{\partial M_{κ, μ} (x)}{\partial κ}$
$- \frac{3}{2}$	$2$	$\begin{matrix} - \frac{4}{x^{3 / 2}} {e^{x / 2} [(x^{3} - 3 x^{2} + 6 x - 6) (Shi (x) - Chi (x) + \ln x + γ) - \frac{11}{6} x^{3} + \frac{15}{2} x^{2} - 15 x + 11] \\ + e^{- x / 2} [6 (Chi (x) + Shi (x) - \ln x - γ) - x^{2} + 4 x - 11]} \end{matrix}$
$- 1$	$\frac{3}{2}$	$\begin{matrix} \frac{3}{2 x} {e^{x / 2} [(2 x^{2} - 4 x + 4) (Chi (x) - Shi (x) - \ln x - γ) + 3 x^{2} - 8 x + 6] \\ + 2 e^{- x / 2} [2 Chi (x) + 2 Shi (x) + x - 2 \ln x - 2 γ - 3]} \end{matrix}$
$- \frac{1}{2}$	$1$	$\begin{matrix} \frac{2}{\sqrt{x}} {e^{x / 2} (x - 1) (Chi (x) - Shi (x) - \ln x - γ + 1) \\ + e^{- x / 2} (\ln x - Chi (x) - Shi (x) + γ + 1)} \end{matrix}$
$- \frac{1}{2}$	$2$	$\begin{matrix} \frac{6}{x^{3 / 2}} {e^{x / 2} [(x^{2} - 4 x + 6) (2 Chi (x) - 2 Shi (x) - 2 \ln x - 2 γ + 3) - 12] \\ + e^{- x / 2} [6 (x - 1) - 4 (x + 3) (\ln x - Chi (x) - Shi (x) + γ)]} \end{matrix}$
$0$	$\frac{3}{2}$	$\begin{matrix} \frac{6}{x} {e^{x / 2} [(x - 2) (Chi (x) - Shi (x) - \ln x - γ) + x] \\ + e^{- x / 2} [(x + 2) (\ln x - Chi (x) - Shi (x) + γ) - x]} \end{matrix}$
$\frac{1}{2}$	$2$	$\begin{matrix} \frac{6}{x^{3 / 2}} {e^{x / 2} [6 (x + 1) - 4 (x - 3) (\ln x + Shi (x) - Chi (x) + γ)] \\ + e^{- x / 2} [(x^{2} + 4 x + 6) (2 \ln x - 2 Chi (x) - 2 Shi (x) + 2 γ - 3) + 12]} \end{matrix}$

Table 7. Table 7: Derivative of M κ , μ subscript 𝑀 𝜅 𝜇 M_{\kappa,\mu} with respect to μ 𝜇 \mu by using ( 24 ).

$κ$	$μ$	$\frac{\partial M_{κ, μ} (x)}{\partial μ}$
$- \frac{3}{2}$	$2$	$\begin{matrix} \frac{4}{x^{3 / 2}} {e^{x / 2} [(x^{3} - 3 x^{2} + 6 x - 6) (Chi (x) - Shi (x) - γ) + \frac{7}{3} x^{3} - 11 x^{2} + 28 x - 36] \\ + e^{- x / 2} [6 (Chi (x) + Shi (x) - γ) + x^{2} - 4 x + 36]} \end{matrix}$
$- 1$	$\frac{3}{2}$	$\begin{matrix} \frac{1}{x} {e^{x / 2} [3 (x^{2} - 2 x + 2) (Chi (x) - Shi (x) - γ) + \frac{13}{2} x^{2} - 22 x + 31] \\ + e^{- x / 2} [3 (x - 2 Chi (x) - 2 Shi (x) + 2 γ) - 31]} \end{matrix}$
$- \frac{1}{2}$	$1$	$\begin{matrix} \frac{2}{\sqrt{x}} {e^{x / 2} [(x - 1) (Chi (x) - Shi (x) - γ + 2) - 2] \\ + e^{- x / 2} (Chi (x) + Shi (x) - γ + 4)} \end{matrix}$
$- \frac{1}{2}$	$2$	$\begin{matrix} \frac{8}{x^{3 / 2}} {e^{x / 2} [3 (\frac{1}{2} x^{2} - 2 x + 3) (Chi (x) - Shi (x) - γ) + 4 x^{2} - 22 x + 48] \\ - e^{- x / 2} [3 (x + 3) (Chi (x) + Shi (x) - γ) + 8 (x + 6)]} \end{matrix}$
$\frac{1}{2}$	$1$	$\begin{matrix} \frac{2}{\sqrt{x}} {e^{x / 2} (Chi (x) - Shi (x) - γ + 4) \\ - e^{- x / 2} [(x + 1) (Chi (x) + Shi (x) - γ + 2) + 2]} \end{matrix}$
$\frac{1}{2}$	$2$	$\begin{matrix} \frac{4}{x^{3 / 2}} {e^{x / 2} [6 (x - 3) (Chi (x) - Shi (x) - γ) + 16 (x - 6)] \\ + e^{- x / 2} [3 (x^{2} + 4 x + 6) (Chi (x) + Shi (x) - γ) + 8 x^{2} + 44 x + 96]} \end{matrix}$

Table 8. Table 8: Integral J 1 ( κ , μ ; x ) subscript 𝐽 1 𝜅 𝜇 𝑥 J_{1}(\kappa,\mu;x) for particular values of κ 𝜅 \kappa and μ 𝜇 \mu .

$κ$	$μ$	$J_{1} (κ, μ; x)$
$- 1$	$0$	$π {2 e^{x / 2} (\ln 4 - 2) [(x + 1) I_{0} (\frac{x}{2}) + x I_{1} (\frac{x}{2})] - G^{(1)} (\frac{3}{2}; 1; x) - 2 H^{(1)} (\frac{3}{2}; 1; x)}$
$- \frac{1}{2}$	$1$	$x^{- 2} {e^{x} [(x - 1) (Chi (x) - Shi (x) - \ln x - γ) - 2] + Chi (x) + Shi (x) - \ln x - γ + 2}$
$- \frac{1}{3}$	$0$	$2 π {G^{(1)} (\frac{5}{6}; 1; x) + 2 H^{(1)} (\frac{5}{6}; 1; x) - \ln (432) L_{- 5 / 6} (x)}$
$0$	$0$	$- π e^{x / 2} {K_{0} (\frac{\| x \|}{2}) + [\ln (4 \| x \|) + γ] I_{0} (\frac{\| x \|}{2})}$
$0$	$\frac{1}{2}$	$x^{- 1} {e^{x} [Chi (x) - Shi (x) - \ln x - γ] - Chi (x) - Shi (x) + \ln x + γ}$
$0$	$1$	$\begin{matrix} {I_{1} (\frac{\| x \|}{2}) [I_{1} (\frac{\| x \|}{2}) K_{1} (\frac{\| x \|}{2}) - \ln (4 \| x \|) - γ + 2 - \frac{1}{2 \sqrt{π}} G_{1, 3}^{2, 1} (\frac{x^{2}}{4}; 1 / 2; 0, 0, - 1)] \\ + K_{1} (\frac{\| x \|}{2}) [1 - I_{0}^{2} (\frac{\| x \|}{2})]} \frac{π}{2 \| x \|} e^{x / 2} \end{matrix}$
$\frac{1}{3}$	$0$	$2 π {G^{(1)} (\frac{1}{6}; 1; x) + 2 H^{(1)} (\frac{1}{6}; 1; x) - \ln (432) L_{- 1 / 6} (x)}$
$\frac{1}{2}$	$\frac{1}{2}$	$\frac{π}{2} {G^{(1)} (\frac{1}{2}; 2; x) + 2 H^{(1)} (\frac{1}{2}; 2; x) - 2 e^{x / 2} \ln 4 [I_{0} (\frac{x}{2}) - I_{1} (\frac{x}{2})]}$
$\frac{1}{2}$	$1$	$x^{- 2} {e^{x} [Chi (x) - Shi (x) - \ln x - γ + 2] - (x + 1) [Chi (x) + Shi (x) - \ln x - γ] - 2}$

Table 9. Table 9: Whittaker function M κ , μ ( x ) subscript 𝑀 𝜅 𝜇 𝑥 M_{\kappa,\mu}(x) for particular values of κ 𝜅 \kappa and μ 𝜇 \mu .

$κ$	$μ$	$M_{κ, μ} (x)$
$- \frac{1}{4}$	$\frac{1}{4}$	$\frac{\sqrt{π}}{2} e^{x / 2} x^{1 / 4} \erf (\sqrt{x})$
$- \frac{1}{2}$	$\frac{1}{2}$	$x [I_{0} (\frac{x}{2}) + I_{1} (\frac{x}{2})]$
$- \frac{1}{2}$	$\frac{1}{6}$	$2^{- 2 / 3} x Γ (\frac{2}{3}) [I_{- 1 / 3} (\frac{x}{2}) + I_{2 / 3} (\frac{x}{2})]$
$- \frac{1}{2}$	$1$	$x^{- 1 / 2} e^{- x / 2} [2 e^{x} (x - 1) + 2]$
$0$	$0$	$\sqrt{x} I_{0} (\frac{x}{2})$
$0$	$\frac{1}{2}$	$2 \sinh (\frac{x}{2})$
$0$	$1$	$4 \sqrt{x} I_{1} (\frac{x}{2})$
$0$	$\frac{3}{2}$	$12 [\cosh (\frac{x}{2}) - \frac{2}{x} \sinh (\frac{x}{2})]$
$0$	$\frac{5}{2}$	$120 x^{- 2} [(x^{2} + 12) \sinh (\frac{x}{2}) - 6 x \cosh (\frac{x}{2})]$
$\frac{1}{6}$	$0$	$\sqrt{x} e^{- x / 2} L_{- 1 / 3} (x)$
$\frac{1}{4}$	$- \frac{1}{4}$	$x^{1 / 4} e^{- x / 2}$
$\frac{1}{4}$	$\frac{1}{4}$	$x^{1 / 4} e^{x / 2} F (\sqrt{x})$
$\frac{1}{3}$	$0$	$\sqrt{x} e^{- x / 2} L_{- 1 / 6} (x)$
$\frac{1}{2}$	$\frac{1}{6}$	$2^{- 2 / 3} x Γ (\frac{2}{3}) [I_{- 1 / 3} (\frac{x}{2}) - I_{2 / 3} (\frac{x}{2})]$
$\frac{1}{2}$	$\frac{1}{4}$	$2^{- 1 / 2} x Γ (\frac{3}{4}) [I_{- 1 / 4} (\frac{x}{2}) - I_{3 / 4} (\frac{x}{2})]$
$\frac{1}{2}$	$\frac{1}{2}$	$x [I_{0} (\frac{x}{2}) - I_{1} (\frac{x}{2})]$
$\frac{1}{2}$	$1$	$2 x^{- 1 / 2} e^{- x / 2} (e^{x} - x - 1)$
$\frac{1}{2}$	$2$	$12 x^{- 3 / 2} e^{- x / 2} [2 e^{x} (x - 3) + x^{2} + 4 x + 6]$
$1$	$- \frac{3}{2}$	$e^{- x / 2} (\frac{x}{2} + 1 + \frac{1}{x})$
$1$	$1$	$\frac{4}{3} \sqrt{x} [x I_{0} (\frac{x}{2}) - (x + 1) I_{1} (\frac{x}{2})]$
$1$	$\frac{3}{2}$	$x^{- 1} e^{- x / 2} (6 e^{x} - 3 x^{2} - 6 x - 6)$
$1$	$2$	$\frac{32}{5} x^{- 1 / 2} [(x^{2} + 4 x + 12) I_{1} (\frac{x}{2}) - (x^{2} + 3 x) I_{0} (\frac{x}{2})]$
$2$	$2$	$\frac{32}{35} x^{- 1 / 2} [x (2 x^{2} + 2 x + 3) I_{0} (\frac{x}{2}) - 2 (x^{3} + 2 x^{2} + 4 x + 6) I_{1} (\frac{x}{2})]$

Equations314

M_{κ, μ} (x)

M_{κ, μ} (x)

2 μ

W_{κ, μ} (x) = \frac{Γ ( - 2 μ )}{Γ ( \frac{1}{2} - μ - κ )} M_{κ, μ} (x) + \frac{Γ ( 2 μ )}{Γ ( \frac{1}{2} + μ - κ )} M_{κ, - μ} (x),

W_{κ, μ} (x) = \frac{Γ ( - 2 μ )}{Γ ( \frac{1}{2} - μ - κ )} M_{κ, μ} (x) + \frac{Γ ( 2 μ )}{Γ ( \frac{1}{2} + μ - κ )} M_{κ, - μ} (x),

{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\Gamma\left(b\right)}{\Gamma\left(a\right)}\sum_{n=0}^{\infty}\frac{\Gamma\left(a+n\right)}{\Gamma\left(b+n\right)}\frac{x^{n}}{n!},

{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\Gamma\left(b\right)}{\Gamma\left(a\right)}\sum_{n=0}^{\infty}\frac{\Gamma\left(a+n\right)}{\Gamma\left(b+n\right)}\frac{x^{n}}{n!},

Mi_{κ, μ} (x)

Mi_{κ, μ} (x)

mi_{κ, μ} (x)

G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\partial}{\partial a}\,\left[{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)\right],

G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\partial}{\partial a}\,\left[{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)\right],

H^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\partial}{\partial b}\,\left[{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)\right].

H^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\partial}{\partial b}\,\left[{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)\right].

\displaystyle G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

\displaystyle G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

\displaystyle H^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\Gamma\left(b\right)}{\Gamma\left(a\right)\Gamma\left(b-a\right)}\int_{0}^{1}e^{xt}t^{a-1}\left(1-t\right)^{b-a-1}dt

{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=\frac{\Gamma\left(b\right)}{\Gamma\left(a\right)\Gamma\left(b-a\right)}\int_{0}^{1}e^{xt}t^{a-1}\left(1-t\right)^{b-a-1}dt

Re b > Re a > 0,

\displaystyle G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

\displaystyle G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

\displaystyle H^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

\displaystyle H^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)

G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=-e^{x}G^{\left(1\right)}\left(\left.\begin{array}[]{c}b-a\\ b\end{array}\right|-x\right).

G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=-e^{x}G^{\left(1\right)}\left(\left.\begin{array}[]{c}b-a\\ b\end{array}\right|-x\right).

{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=e^{x}\,_{1}F_{1}\left(\left.\begin{array}[]{c}b-a\\ b\end{array}\right|-x\right),

{}_{1}F_{1}\left(\left.\begin{array}[]{c}a\\ b\end{array}\right|x\right)=e^{x}\,_{1}F_{1}\left(\left.\begin{array}[]{c}b-a\\ b\end{array}\right|-x\right),

\frac{\partial M _{κ, μ} ( x )}{\partial κ}

\frac{\partial M _{κ, μ} ( x )}{\partial κ}

S_{1} (κ, μ, x) = n = 0 \sum \infty \frac{Γ ( \frac{1}{2} + μ - κ + n )}{Γ ( 1 + 2 μ + n )} ψ (\frac{1}{2} + μ - κ + n) \frac{x ^{n}}{n !} .

S_{1} (κ, μ, x) = n = 0 \sum \infty \frac{Γ ( \frac{1}{2} + μ - κ + n )}{Γ ( 1 + 2 μ + n )} ψ (\frac{1}{2} + μ - κ + n) \frac{x ^{n}}{n !} .

\left.\frac{\partial\mathrm{M}_{\kappa,\mu}\left(x\right)}{\partial\kappa}\right|_{\kappa=-\mu-1/2}=-\frac{x^{\mu+3/2}}{2\mu+1}e^{x/2}\,_{2}F_{2}\left(\left.\begin{array}[]{c}1,1\\ 2\left(\mu+1\right),2\end{array}\right|-x\right).

\left.\frac{\partial\mathrm{M}_{\kappa,\mu}\left(x\right)}{\partial\kappa}\right|_{\kappa=-\mu-1/2}=-\frac{x^{\mu+3/2}}{2\mu+1}e^{x/2}\,_{2}F_{2}\left(\left.\begin{array}[]{c}1,1\\ 2\left(\mu+1\right),2\end{array}\right|-x\right).

\frac{\partial M _{κ, μ} ( x )}{\partial κ}_{κ = - μ - 1/2}

\frac{\partial M _{κ, μ} ( x )}{\partial κ}_{κ = - μ - 1/2}

\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\psi\left(k+a\right)=e^{t}\left[\psi\left(a\right)+\frac{t}{a}\,_{2}F_{2}\left(\left.\begin{array}[]{c}1,1\\ a+1,2\end{array}\right|-t\right)\right].

\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\psi\left(k+a\right)=e^{t}\left[\psi\left(a\right)+\frac{t}{a}\,_{2}F_{2}\left(\left.\begin{array}[]{c}1,1\\ a+1,2\end{array}\right|-t\right)\right].

G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ a\end{array}\right|x\right)=\frac{x\,e^{x}}{a}\,_{2}F_{2}\left(\left.\begin{array}[]{c}1,1\\ a+1,2\end{array}\right|-x\right).

G^{\left(1\right)}\left(\left.\begin{array}[]{c}a\\ a\end{array}\right|x\right)=\frac{x\,e^{x}}{a}\,_{2}F_{2}\left(\left.\begin{array}[]{c}1,1\\ a+1,2\end{array}\right|-x\right).

\frac{\partial\mathrm{M}_{\kappa,\mu}\left(x\right)}{\partial\kappa}=-x^{\mu+1/2}e^{-x/2}G^{\left(1\right)}\left(\left.\begin{array}[]{c}\frac{1}{2}+\mu-\kappa\\ 1+2\mu\end{array}\right|x\right),

\frac{\partial\mathrm{M}_{\kappa,\mu}\left(x\right)}{\partial\kappa}=-x^{\mu+1/2}e^{-x/2}G^{\left(1\right)}\left(\left.\begin{array}[]{c}\frac{1}{2}+\mu-\kappa\\ 1+2\mu\end{array}\right|x\right),

\frac{\partial M _{κ, μ} ( x )}{\partial κ}_{κ = n, μ = 1/2}

\frac{\partial M _{κ, μ} ( x )}{\partial κ}_{κ = n, μ = 1/2}

a_{ℓ} = \frac{1}{n} (\frac{n + ℓ}{n - ℓ}),

a_{ℓ} = \frac{1}{n} (\frac{n + ℓ}{n - ℓ}),

b_{\ell}=\left\{\begin{array}[]{ll}\displaystyle\frac{1}{n}\sum_{k=0}^{n-\ell-1}\frac{\left(\ell\right)_{k}2^{k}}{\left(\ell+n\right)_{k}},&\ell=1,2,\ldots\\ 0,&\ell=0.\end{array}\right.

b_{\ell}=\left\{\begin{array}[]{ll}\displaystyle\frac{1}{n}\sum_{k=0}^{n-\ell-1}\frac{\left(\ell\right)_{k}2^{k}}{\left(\ell+n\right)_{k}},&\ell=1,2,\ldots\\ 0,&\ell=0.\end{array}\right.

S\left(n,\ell\right)=\sum_{k=0}^{n-\ell-1}\frac{\left(\ell\right)_{k}2^{k}}{\left(\ell+n\right)_{k}}=\mathrm{Re}\left[{}_{2}F_{1}\left(\left.\begin{array}[]{c}1,\ell\\ \ell+n\end{array}\right|2\right)\right].

S\left(n,\ell\right)=\sum_{k=0}^{n-\ell-1}\frac{\left(\ell\right)_{k}2^{k}}{\left(\ell+n\right)_{k}}=\mathrm{Re}\left[{}_{2}F_{1}\left(\left.\begin{array}[]{c}1,\ell\\ \ell+n\end{array}\right|2\right)\right].

S (n, ℓ) = S_{1} (n, ℓ) k = 0 \sum \infty \frac{( ℓ ) _{k} ( 1 ) _{k} 2 ^{k}}{k ! ( ℓ + n ) _{k}} - S_{2} (n, ℓ) k = n - ℓ \sum \infty \frac{( ℓ ) _{k} ( 1 ) _{k} 2 ^{k}}{k ! ( ℓ + n ) _{k}},

S (n, ℓ) = S_{1} (n, ℓ) k = 0 \sum \infty \frac{( ℓ ) _{k} ( 1 ) _{k} 2 ^{k}}{k ! ( ℓ + n ) _{k}} - S_{2} (n, ℓ) k = n - ℓ \sum \infty \frac{( ℓ ) _{k} ( 1 ) _{k} 2 ^{k}}{k ! ( ℓ + n ) _{k}},

S_{1}\left(n,\ell\right)=\,_{2}F_{1}\left(\left.\begin{array}[]{c}1,\ell\\ \ell+n\end{array}\right|2\right),

S_{1}\left(n,\ell\right)=\,_{2}F_{1}\left(\left.\begin{array}[]{c}1,\ell\\ \ell+n\end{array}\right|2\right),

S_{2} (n, ℓ)

S_{2} (n, ℓ)

{}_{2}F_{1}\left(\left.\begin{array}[]{c}a,b\\ 2b\end{array}\right|z\right)

{}_{2}F_{1}\left(\left.\begin{array}[]{c}a,b\\ 2b\end{array}\right|z\right)

{}_{2}F_{1}\left(\left.\begin{array}[]{c}1,n\\ 2n\end{array}\right|2\right)=i\,_{2}F_{1}\left(\left.\begin{array}[]{c}\frac{1}{2},n-\frac{1}{2}\\ n+\frac{1}{2}\end{array}\right|1\right).

{}_{2}F_{1}\left(\left.\begin{array}[]{c}1,n\\ 2n\end{array}\right|2\right)=i\,_{2}F_{1}\left(\left.\begin{array}[]{c}\frac{1}{2},n-\frac{1}{2}\\ n+\frac{1}{2}\end{array}\right|1\right).

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsMathematical functions and polynomials

Full text

Part 1. Infinite series and logarithmic integrals associated to

differentiation with respect to parameters of the Whittaker $\mathrm{M}_{\kappa,\mu}\left(x\right)$ function.

Alexander Apelblat1, Juan Luis González-Santander2.

1 Department of Chemical Engineering,

Ben Gurion University of the Negev,

84105 Beer Sheva, 84105, Israel. [email protected]

2 Department of Mathematics, Universidad de Oviedo,

33007 Oviedo, Spain. [email protected]

Abstract

First derivatives of the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ . These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed-form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function has been derived, as well as some finite and infinite integrals containing products of algebraic, exponential, logarithmic and Bessel functions. Finally, some reduction formulas for the Whittaker functions $\mathrm{M}_{\kappa,\mu}\left(x\right)$ and integral Whittaker functions $\mathrm{Mi}_{\kappa,\mu}\left(x\right)$ and $\mathrm{mi}_{\kappa,\mu}\left(x\right)$ are calculated.

Keywords: Derivatives with respect to parameters; Whittaker functions; integral Whittaker functions; incomplete gamma functions; sums of infinite series of psi and gamma; finite and infinite logarithmic integrals and Bessel functions.

AMS Subject Classification: 33B15, 33B20, 33C10, 33C15, 33C20, 33C50, 33E20.

1 Introduction

Introduced in 1903 by Whittaker [30], the $\mathrm{M}_{\kappa,\mu}\left(x\right)$ and $\mathrm{W}_{\kappa,\mu}\left(x\right)$ functions are defined as:

[TABLE]

and

[TABLE]

where $\Gamma\left(x\right)$ denotes the gamma function. These functions, called Whittaker functions, are closely associated to the following confluent hypergeometric function (Kummer function):

[TABLE]

where ${}_{p}F_{q}\left(\left.\begin{array}[]{c}a_{1},\ldots,a_{p}\\ b_{1},\ldots,b_{q}\end{array}\right|x\right)$ denotes the generalized hypergeometric function.

For particular values of the parameters $\kappa$ and $\mu$ , the Whittaker functions can be reduced to a variety of elementary and special functions. Whittaker [30] discussed the connection of the functions defined in (3) and (4) with many other special functions, such as the modified Bessel function, the incomplete gamma functions, the parabolic cylinder function, the error functions, the logarithmic and the cosine integrals, and the generalize Hermite and Laguerre polynomials. Monographs and treatises dealing with special functions [13, 28, 31, 24, 22, 10, 16, 26, 23] present the properties of the Whittaker functions with more or less extension.

The Whittaker functions are frequently applied in various areas of mathematical physics (see, for example [12, 27, 25]), such as the well-known solution of the Schrödinger equation for the harmonic oscillator [18].

$\mathrm{M}_{\kappa,\mu}\left(x\right)$ and $\mathrm{W}_{\kappa,\mu}\left(x\right)$ are usually treated as functions of variable $x$ with fixed values of the parameters $\kappa$ and $\mu$ . However, there are few investigations which consider $\kappa$ and $\mu$ as variables. For instance, Laurenzi [19] discussed methods to calculate derivatives of $\mathrm{M}_{\kappa,1/2}\left(x\right)$ and $\mathrm{W}_{\kappa,1/2}\left(x\right)$ with respect to $\kappa$ when this parameter is an integer. Using the Mellin transform, Buschman [11] showed that the derivatives of certain Whittaker functions with respect to the parameters can be expressed in finite sums of Whittaker functions. López and Sesma [21] considered the behaviour of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ as a function of $\kappa$ . They derived a convergent expansion in ascending powers of $\kappa$ , and an asymptotic expansion in descending powers of $\kappa$ . Using series of Bessel functions and Buchholz polynomials, Abad and Sesma [1] presented an algorithm for the calculation of the $n$ -th derivative of the Whittaker functions with respect to parameter $\kappa$ . Becker [6] investigated certain integrals with respect to parameter $\mu$ . Ancarini and Gasaneo [2] presented a general case of differentiation of generalized hypergeometric functions with respect to the parameters in terms of infinite series containing the digamma function. In addition, Sofostasios and Brychkov [29] considered derivatives of hypergeometric functions and classical polynomials with respect to the parameters.

In this paper, our main focus will be directed to the systematic investigation of the first derivatives of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to the parameters. We will mainly base our results on two different approaches. The first one has to do with the series representation of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ , and the second one has to do with the integral representations of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ . Regarding the first approach, direct differentiation of (3) with respect to the parameters leads to infinite sums of quotients of digamma and gamma functions. It is possible to calculate such sums in closed-form for particular values of the parameters. The parameter differentiation of the integral representations of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ leads to finite and infinite integrals of elementary functions, such as products of algebraic functions, exponential and logarithmic functions. These integrals are similar to those investigated by Kölbig [17] and Geddes et al. [14]. As in the case of the first approach, it is possible to calculate such integrals in closed-form for some particular values of the parameters.

In the Appendices, we calculate the first derivative of the incomplete gamma functions $\gamma\left(\nu,x\right)$ and $\Gamma\left(\nu,x\right)$ with respect to the parameter $\nu$ . These results will be used when we calculate some of the integrals found in the second approach mentioned before. Also, we calculate some new reduction formulas of the integral Whittaker functions, which were recently introduced by us in [4]. They are defined in a similar way as other integral functions in the mathematical literature:

[TABLE]

Finally, we also include a list of reduction formulas for the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ in the Appendices.

2 Parameter differentiation of $\mathrm{M}_{\kappa,\mu}$ via Kummer function ${}_{1}F_{1}$

As mentioned before, the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ is closely related to the confluent hypergeometric function ${}_{1}F_{1}\left(a;b;x\right)$ . Likewise, the parameter derivatives of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ are also related to the parameter derivatives of ${}_{1}F_{1}\left(a;b;x\right)$ . Let us introduce the following notation set by Ancarini and Gasaneo [2].

Definition 1

Define the parameter derivatives of the confluent hypergeometric function as,

[TABLE]

and

[TABLE]

According to (5), we have

[TABLE]

Since one of the integral representations of the confluent hypergeometric function is [22, Sect. 6.5.1]:

[TABLE]

by direct differentiation of (14) with respect to parameters $a$ and $b$ , we obtain

[TABLE]

and

[TABLE]

Since the main focus is the systematic investigation of the parameter derivatives of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ , we will present these parameter derivatives as Theorems along the paper, and the corresponding results for $G^{\left(1\right)}\left(a;b;x\right)$ and $H^{\left(1\right)}\left(a;b;x\right)$ as Corollaries. Also, note that all the results regarding $G^{\left(1\right)}\left(a;b;x\right)$ can be transformed according to the next Theorem.

Theorem 2

The following transformation holds true:

[TABLE]

Proof. Differentiate with respect to $a$ Kummer’s transformation formula [24, Eqn. 13.2.39]:

[TABLE]

to obtain the desired result.

2.1 Derivative with respect to the first parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa$

Using (3) and (5), the first derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to the first parameter $\kappa$ is

[TABLE]

where $\psi\left(x\right)$ denotes the digamma function and

[TABLE]

Theorem 3

For $\mu\neq-1/2$ and for $x\in\mathbb{R}$ , the following parameter derivative formula of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ holds true:

[TABLE]

Proof. For $\kappa=-\mu-1/2$ , Eqn. (2.1) becomes

[TABLE]

Apply [9, Eqn. 6.2.1(60)]

[TABLE]

to obtain (19), as we wanted to prove.

Corollary 4

For $a\in\mathbb{R}$ , $a\neq 0$ , and for $x\in\mathbb{R}$ , the following reduction formula holds true:

[TABLE]

Proof. Direct differentiation of (3) yields

[TABLE]

thus comparing (22) with $\kappa=-\mu-\frac{1}{2}$ to (19), we arrive at (21), as we wanted to prove.

Table 1 presents some explicit expressions for particular values of (19), and for $x\in\mathbb{R}$ , obtained with the help of MATHEMATICA program.

Next, we present other reduction formula of $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa$ from the result found in [19] for $x\in\mathbb{R}$ ,

[TABLE]

where $\mathrm{Ei}\left(x\right)$ denotes the exponential integral and for $n,\ell=1,2,\ldots$

[TABLE]

and

[TABLE]

In order to calculate the finite sum given in (25), we derive the following Lemma.

Lemma 5

The following finite sum holds true $\forall n,\ell=1,2,\ldots$

[TABLE]

Proof. Split the sum in two as

[TABLE]

where

[TABLE]

and

[TABLE]

Take $a=1$ , $b=n$ , and $z=2$ in the quadratic transformation [24, Eqn. 15.18.3]

[TABLE]

to obtain

[TABLE]

Now, apply Gauss’s summation theorem [24, Eqn. 15.4.20]

[TABLE]

and the formula [23, Eqn. 43:4:3]

[TABLE]

to arrive at

[TABLE]

Therefore, $S_{2}\left(n,\ell\right)$ is a pure imaginary number. Since $S\left(n,\ell\right)$ is a real number, we conclude that $S\left(n,\ell\right)=\mathrm{Re}\left[S_{1}\left(n,\ell\right)\right]$ , as we wanted to prove.

Theorem 6

The following reduction formula holds true for $n=1,2,\ldots$ and $x\in\mathbb{R}$ ,

[TABLE]

where $L_{n}^{\left(\alpha\right)}\left(x\right)$ denotes the Laguerre polynomials (182) and $H_{n}=\sum_{k=1}^{n}\frac{1}{k}$ the $n$ -th harmonic number.

Proof. From (26) and (25), we see that

[TABLE]

Also, according to [24, Eqn. 13.18.1]

[TABLE]

In addition, performing the transformations $\kappa\rightarrow\kappa+1$ , $\kappa\rightarrow 0,$ and $n\rightarrow n-1$ in (181), we obtain $\forall n=1,2,\ldots$

[TABLE]

Finally, we have for $n=1,2,\ldots$ [20, Eqn. 1.3.7]

[TABLE]

Insert (24) and (25)-(38) in (2.1) to arrive at (34), as we wanted to prove.

Corollary 7

The following reduction formula holds true for $n=1,2,\ldots$ and $x\in\mathbb{R}$ ,

[TABLE]

Proof. Consider (22) and (34) to arrive at the desired result.

In Table 2 we collect some particular cases of (34) for $x\in\mathbb{R}$ obtained with the help of MATHEMATICA program.

2.2 Derivative with respect to the second parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\mu$

Using (3) and (5), the first derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to the parameter $\mu$ is

[TABLE]

where $S_{1}\left(\kappa,\mu,x\right)$ is given in (18) and the series $S_{2}\left(\kappa,\mu,x\right)$ is

[TABLE]

Theorem 8

For $\mu\neq-1/2$ and $x\in\mathbb{R}$ , the following parameter derivative formula of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ holds true:

[TABLE]

Proof. For $\kappa=-\mu-1/2$ , we have $S_{2}\left(\kappa,\mu,x\right)=2\,S_{1}\left(\kappa,\mu,x\right)$ and therefore (2.2) becomes

[TABLE]

where

[TABLE]

thus, using (20),

[TABLE]

Since, according to (3) and (5),

[TABLE]

then (50) takes the simple form given in (43), as we wanted to prove.

Corollary 9

For $a\in\mathbb{R}$ , $a\neq 0$ , and $x\in\mathbb{R}$ , the following reduction formula holds true:

[TABLE]

Proof. Direct differentiation of (3) yields

[TABLE]

thus comparing (57) with $\kappa=-\mu-\frac{1}{2}$ to (43) and taking into account (21), we arrive at (51), as we wanted to prove.

Using (43), the derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect $\mu$ has been calculated for particular values of $\kappa$ and $\mu$ , with $x\in\mathbb{R}$ , using the help of MATHEMATICA, and they are presented in Table 3.

Note that for $\mu=-1/2$ , we obtain an indeterminate expression in (43). For this case, we present the following result.

Theorem 10

The following parameter derivative formula of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ holds true for $x\in\mathbb{R}$ :

[TABLE]

where $I_{\nu}\left(x\right)$ denotes the modified Bessel function.

Proof. Differentiate with respect to $\mu$ the expression [24, Eqn. 13.18.8]

[TABLE]

to obtain (10), as we wanted to prove.

The order derivative of the modified Bessel function $I_{\mu}\left(x\right)$ is given in terms of the Meijer-G function and the generalized hypergeometric function $\forall\,\mathrm{Re\,}x>0,\mu\geq 0$ [15]:

[TABLE]

where $K_{\nu}\left(x\right)$ is the modified Bessel function of the second kind; or in terms of generalized hypergeometric functions only $\forall\,\mathrm{Re\,}x>0,\mu>0$ , $\mu\notin\mathbb{Z}$ [7]:

[TABLE]

There are different expressions for the order derivatives of the Bessel functions [5, 8]. This subject is summarized in [3], where general results are presented in terms of convolution integrals, and order derivatives of Bessel functions are found for particular values of the order.

Using (10), (65) and (66), some derivatives of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect $\mu$ has been calculated for $x\in\mathbb{R}$ with the help of MATHEMATICA, and they are presented in Table 4.

3 Parameter differentiation of $\mathrm{M}_{\kappa,\mu}$ via integral representations

3.1 Derivative with respect to the first parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa$

Integral representations of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ can be obtained via integral representations of confluent hypergeometric function [22, Sect. 7.4.1], thus

[TABLE]

where

[TABLE]

denotes the beta function. In order to calculate the first derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to parameter $\kappa$ , let us introduce the following finite logarithmic integrals.

Definition 11

[TABLE]

Differentiation of (73) and (3.1) with respect to parameter $\kappa$ yields respectively

[TABLE]

Note that, from (3.1) and (3.1), we have

[TABLE]

Likewise, we can depart from other integral respresentations of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ [22, Sect. 7.4.1]111There are some typos in this reference regarding these integral representations., to obtain

[TABLE]

and consequently, we have

[TABLE]

where we have defined the following logarithmic integrals:

Definition 12

[TABLE]

Note that, from (3.1)-(3.1), we have

[TABLE]

Since $\,I_{2}\left(\kappa,\mu;x\right)$ , $\,I_{3}\left(\kappa,\mu;x\right)$ , and $\,I_{4}\left(\kappa,\mu;x\right)$ are reduced to the calculation of $\,I_{1}\left(\kappa,\mu;x\right)$ , next we calculate the latter integral.

Theorem 13

The following integral holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. Comparing (3.1) to (22), taking into account (3), we arrive at (94), as we wanted to prove.

Corollary 14

For $\kappa=0$ , Eqn. (94) is reduced to

[TABLE]

Theorem 15

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following integral holds true for $x\in\mathbb{R}$ :

[TABLE]

where

[TABLE]

and the functions $\mathrm{Shi}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ denote the hyperbolic sine and cosine integrals.

Proof. From the definition of $I_{1}\left(\kappa,\mu;x\right)$ given in (76), we have

[TABLE]

Perform the change of variables $\tau=1-t$ in the first integral above to arrive at

[TABLE]

where we have set

[TABLE]

Taking into account the binomial theorem and the integral (177) calculated in the Appendix, i.e.

[TABLE]

calculate

[TABLE]

Now, apply the differentiation formula [24, Eqn. 16.3.1]

[TABLE]

to obtain

[TABLE]

According to [26, Eqn. 7.12.2(67)], we have that

[TABLE]

In order to obtain similar expressions as the ones obtained in Table 1, we derive an alternative form of (105). Indeed, from the definition of the hyperbolic sine and cosine integrals [24, Eqns. 6.2.15-16], $\forall z\in\mathbb{C}$ ,

[TABLE]

it is easy to prove that

[TABLE]

Also, from the definition of complementary exponential integral [24, Eqn. 6.2.3]:

[TABLE]

and the property $\forall x>0$ [24, Eqn. 6.2.7]

[TABLE]

it is easy to prove that

[TABLE]

thus, taking into account (106) and (107), we have

[TABLE]

Insert (108) in (105), to obtain

[TABLE]

Finally, substitute (109) in (104), and take into account (15), to arrive at

[TABLE]

Similarly, calculate

[TABLE]

Finally, according to (98), we arrive at (96), as we wanted to prove.

Table 5 shows the integral $I_{1}\left(\kappa,\mu;x\right)$ for $x\in\mathbb{R}$ and particular values of the parameters $\kappa$ and/or $\mu$ , obtained from (94) and (96) with the aid of MATHEMATICA program.

Theorem 16

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following reduction formula holds true for $x\in\mathbb{R}$ :

[TABLE]

where we have set the polynomials:

[TABLE]

Proof. According to the definition of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ (3), we have

[TABLE]

Applying the property [23, Eqn. 18:5:1]

[TABLE]

and the reduction formula [26, Eqn. 7.11.1(12)]

[TABLE]

where $n,m=1,2,\ldots$ and $m>n$ , after some algebra, we arrive at

[TABLE]

Insert (3.1) in (113) to obtain (16), as we wanted to prove.

In addition to (16), other reduction formulas for the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ are presented in Appendix C. A large list of reduction formulas for $\mathrm{M}_{\kappa,\mu}\left(x\right)$ is available in [4] and in other monographs dealing with the special functions [13, 28, 31, 24, 22, 10, 16, 26, 23, 9].

Theorem 17

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following reduction formula holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. According to (3.1), we have

[TABLE]

Now, apply (75) and the property (38) to get

[TABLE]

Finally, applying the results given in (96) and (16), we arrive at (17), as we wanted to prove.

Corollary 18

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following reduction formula holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. Set (22) for $\kappa=\frac{\ell}{2}$ and $\mu=m+\frac{1-\ell}{2}$ and compare the result to (17).

Table 6 shows the first derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to parameter $\kappa$ for some particular values of $\kappa$ and $\mu$ , and $x\in\mathbb{R}$ , which has been calculated from (17) and are not contained in Table 1.

3.2 Application to the calculation of infinite integrals

Additional integral representations of the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ in terms of Bessel functions [22, Sect. 6.5.1] are known:

[TABLE]

Let us introduce the following infinite logarithmic integrals.

Definition 19

[TABLE]

Differentiation of (123) and (3.2) with respect to parameter $\kappa$ yields respectively

[TABLE]

Note that, from (3.1) and (127) we have

[TABLE]

and from (3.1) and (128),

[TABLE]

Corollary 20

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following infinite integrals holds true for $x\in\mathbb{R}$ :

[TABLE]

and

[TABLE]

Proof. Substitute in (3.2) and (3.2) the results given in (96) and (16), and apply (38).

3.3 Derivative with respect to the second parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\mu$

In order to calculate the first derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to parameter $\mu$ , let us introduce the following finite logarithmic integrals.

Definition 21

[TABLE]

Differentiation of (73) and (3.1) with respect to parameter $\mu$ gives

[TABLE]

For the other integral representations given in (3.1) and (3.1), we have

[TABLE]

From (138)-(140), we obtain the following interrelationships:

[TABLE]

Since $\,J_{2}\left(\kappa,\mu;x\right)$ , $\,J_{3}\left(\kappa,\mu;x\right)$ , and $\,J_{4}\left(\kappa,\mu;x\right)$ are reduced to the calculation of $\,J_{1}\left(\kappa,\mu;x\right)$ , next we calculate the latter integral.

Theorem 22

According to the notation introduced in (8) and (9), the following integral holds true:

[TABLE]

Proof. Comparing (138) to (57), taking into account (3), we arrive at (149), as we wanted to prove.

Theorem 23

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following integral holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. From the definition of $J_{1}\left(\kappa,\mu;x\right)$ given in (133), we have

[TABLE]

Perform the change of variables $\tau=1-t$ in the second integral above to arrive at

[TABLE]

where we follow the notation given in (99) for the integral $\mathcal{I}_{1}\left(\kappa,\mu;x\right)$ . According to the results obtained in (3.1) and (110), we arrive at (150), as we wanted to prove.

Theorem 24

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following reduction formula holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. Insert (16) and (150) in (138) and apply (38).

Table 7 shows the first derivative of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ with respect to parameter $\mu$ for some particular values of $\kappa$ and $\mu$ , andfor $x\in\mathbb{R}$ , which has been calculated from (24) and are not contained in Tables 3 and 4.

Corollary 25

For $\ell\in\mathbb{Z}$ and $m=0,1,2,\ldots$ , with $m\geq\ell$ , the following reduction formula holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. Take $\kappa=\frac{\ell}{2}$ and $\mu=m+\frac{1-\ell}{2}$ in (57), and substitute the results given in (16), (18) and (24). After simplification, we arrive at (25), as we wanted to prove.

3.4 Application to the calculation of finite integrals

Theorem 26

For $\mu\geq 0$ and $x\in\mathbb{R}$ , the following finite integral holds true:

[TABLE]

where $\partial I_{\mu}\left(x\right)/\partial\mu$ is given by (65) or (66).

Proof. First, consider that $x>0$ . Take $\kappa=0$ in (138) and susbtitute (59) to arrive at

[TABLE]

Next, equate (3.4) to the expression given in (10), and solve for $J_{1}\left(0,\mu;x\right)$ to get

[TABLE]

Now, apply the property [24, Eqn. 5.5.8]

[TABLE]

for $z=\mu+\frac{1}{2}$ to simplify (3.4) as

[TABLE]

where (3.4) holds true for $x>0$ . Finally, note that performing in (133) the change of variables $\tau=1-t$ , we obtain the reflection formula

[TABLE]

so that from (3.4) and (160) we arrive at (26), as we wanted to prove.

Theorem 27

For $\mu\geq 0$ and $x\in\mathbb{R}$ , the following finite integral holds true:

[TABLE]

where $\partial I_{\mu}\left(x\right)/\partial\mu$ is given by (65) or (66).

Proof. Consider $x>0$ . Take $\kappa=0$ in (140) and susbtitute (59) to obtain

[TABLE]

Now, insert in (3.4) the result given in (3.4) and simplify to get for $x>0$

[TABLE]

Finally, note that performing in (135) the change of variables $\tau=-t$ , we obtain the reflection formula

[TABLE]

so that from (3.4) and (164) we arrive at (27), as we wanted to prove.

Table 8 shows the integral $J_{1}\left(\kappa,\mu;x\right)$ for particular values of the parameters $\kappa$ and $\mu$ , and $x\in\mathbb{R}$ , obtained from (149), (150) and (26) with the aid of MATHEMATICA program.

4 Conclusions

The Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ is defined in terms of the Kummer confluent hypergeometric function, hence its derivative with respect to the parameters $\kappa$ and $\mu$ can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, the parameter differentiation of the integral representations of $\mathrm{M}_{\kappa,\mu}\left(x\right)$ leads to finite and infinite integrals of elementary functions. These sums and integrals has been calculated for particular values of the parameters $\kappa$ and $\mu$ in closed-form. As an application of these results, we have obtained some reduction formulas for the derivatives of the confluent Kummer function with respect to the parameters, i.e. $G^{\left(1\right)}\left(a,b;x\right)$ and $H^{\left(1\right)}\left(a,b;x\right)$ . Also, we have calculated some finite integrals containing a combination of the exponential, logarithmic and algebraic functions, as well as some infinite integrals involving the exponential, logarithmic, algebraic and Bessel functions. It is worth noting that all the results presented in this paper has been both numerically and symbolically checked with MATHEMATICA program.

In the first Appendix, we have obtained the first derivative of the incomplete gamma functions in closed-form. These results allow us to calculate a finite logarithmic integral, which has been used to calculate one of the integrals appearing in the body of the paper.

In the second Appendix, we have calculated some new reduction formulas for the integral Whittaker functions $\mathrm{Mi}_{\kappa,\mu}\left(x\right)$ and $\mathrm{mi}_{\kappa,\mu}\left(x\right)$ from two reduction formulas of the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ . One of the latter seems not to be reported in the literature.

In the third Appendix we collect some reduction formulas for the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ .

Appendix A Parameter differentiation of the incomplete gamma functions

Definition 28

The lower incomplete gamma function is defined as [23]:

[TABLE]

Definition 29

The upper incomplete gamma function is defined as [23, Eqn. 45:3:2]

[TABLE]

The relation between both functions is

[TABLE]

The lower incomplete gamma function has the following series expansion [23, Eqns. 45:6:1]

[TABLE]

Also, the following integral representations in terms of infinite integrals hold true [24, Eqns. 8.6.3&7] for $\mathrm{Re}\,z>0$ ,

[TABLE]

From (165), the derivative of the lower incomplete gamma function with respect to the order $\nu$ has the following integral representation:

[TABLE]

Theorem 30

The parameter derivative of the lower incomplete gamma function is

[TABLE]

Proof. According to (165) and (168), the derivative of the lower incomplete gamma function with respect to the parameter $\nu$ is,

[TABLE]

Now, apply the sum formula [9, Eqn. 6.2.1(63)]

[TABLE]

to arrive at (170), as we wanted to prove.

Theorem 31

The parameter derivative of the upper incomplete gamma function is

[TABLE]

Proof. Differentiate (167) with respect to the parameter $\nu$ and apply the result given in (170).

Corollary 32

From (169) and (170), we calculate the following integral:

[TABLE]

Corollary 33

The following integral holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. Perform the change of variables $t=z\,\tau$ in the integral given in (176), split the result in two integrals and apply again the change of variables $t=x\,\tau$ to the first integral,

[TABLE]

Comparing (176) to (178), we obtain (177), as we wanted to prove.

Corollary 34

According to the notation given in (9), the following reduction formula holds true for $x\in\mathbb{R}$ :

[TABLE]

Proof. Knowing that [23, Eqn. 47:4:6]

[TABLE]

and applying (170), we calculate (179), as we wanted to prove.

Appendix B Reduction formulas for integral Whittaker functions $\mathrm{Mi}_{\kappa,\mu}$ and $\mathrm{mi}_{\kappa,\mu}$

In [4], we found some reduction formulas for the integral Whittaker function $\mathrm{Mi}_{\kappa,\mu}\left(x\right)$ . Next, we derive some new reduction formulas for $\mathrm{Mi}_{\kappa,\mu}\left(x\right)$ and $\mathrm{mi}_{\kappa,\mu}\left(x\right)$ from reduction formulas of the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ .

Theorem 35

The following reduction formula holds true for $x\in\mathbb{R}$ , $n=0,1,2,\ldots$ and $\kappa>0$ :

[TABLE]

where $\gamma\left(\nu,z\right)$ denotes the lower incomplete gamma function.

Proof. Apply to the definition of the Whittaker function (3) the reduction formula [26, Eqn. 7.11.1(17)]

[TABLE]

to obtain [24, Eqn. 13.18.17]

[TABLE]

where [20, Eqn. 4.17.2]

[TABLE]

denotes the Laguerre polynomials. Insert (182) in (181) and integrate term by term according to the definition of the integral Whittaker function (6), to get

[TABLE]

Finally, take into account the defintion of the lower incomplete gamma function (165) and simplify the result to arrive at (180), as we wanted to prove.

Remark 36

Taking $n=0$ in (180), we recover the formula given in [4].

Theorem 37

The following reduction formula holds true for $x>0$ , $n=0,1,2,\ldots$ and $\kappa\in\mathbb{R}$ :

[TABLE]

where $\Gamma\left(\nu,z\right)$ denotes the upper incomplete gamma function.

Proof. Follow similar steps as in the previous theorem, but consider the definition of the upper incomplete gamma function (166).

Theorem 38

The following reduction formula holds true for $x\in\mathbb{R}$ , $n=0,1,2,\ldots$ and $\kappa>0$ :

[TABLE]

Proof. From the property for $x>0$ [23, Eqn. 48:13:3]

[TABLE]

we have, for $x\in\mathbb{R}$ ,

[TABLE]

Apply (185) to (181) to obtain

[TABLE]

Now, insert (182) in (181) and integrate term by term according to the definition of the integral Whittaker function (6), to get

[TABLE]

Finally, take into account the defintion of the lower incomplete gamma function (165) and simplify the result to arrive at (184), as we wanted to prove.

Remark 39

It is worth noting that we could not locate the reduction formula (186) in the literature.

Appendix C Reduction formulas for the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$

For convenience of the readers, reduction formulas for the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$ are presented in their explicit form in Table 9 for $x\in\mathbb{R}$ .

Bibliography31

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] J Abad and J Sesma. Successive derivatives of Whittaker functions with respect to the first parameter. Computer Physics Communications , 156(1):13–21, 2003.
2[2] LU Ancarani and Gustavo Gasaneo. Derivatives of any order of the hypergeometric function pfq(a 1,…, ap; b 1,…, bq; z) with respect to the parameters ai and bi. Journal of Physics A: Mathematical and Theoretical , 43(8):085210, 2010.
3[3] Alexander Apelblat. Bessel and Related Functions: Mathematical Operations with Respect to the Order . De Gruyter, Berlin, 2020.
4[4] Alexander Apelblat and Juan Luis González-Santander. The Integral Mittag-Leffler, Whittaker and Wright functions. Mathematics , 9(24):3255, 2021.
5[5] Alexander Apelblat and Naftali Kravitsky. Integral representations of derivatives and integrals with respect to the order of the Bessel functions j ν ( t ) subscript 𝑗 𝜈 𝑡 j_{\nu}(t) , i ν ( t ) subscript 𝑖 𝜈 𝑡 i_{\nu}(t) , the Anger function 𝐉 ν ( t ) subscript 𝐉 𝜈 𝑡 \mathbf{J}_{\nu}(t) and the integral Bessel function j i ν ( t ) 𝑗 subscript 𝑖 𝜈 𝑡 ji_{\nu}(t) . IMA journal of applied mathematics , 34(2):187–210, 1985.
6[6] Peter A Becker. Infinite integrals of Whittaker and Bessel functions with respect to their indices. Journal of mathematical physics , 50(12):123515, 2009.
7[7] Yu A Brychkov. Higher derivatives of the Bessel functions with respect to the order. Integral Transforms and Special Functions , 27(7):566–577, 2016.
8[8] Yu A Brychkov and KO Geddes. On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms and Special Functions , 16(3):187–198, 2005.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

Part 1. Infinite series and logarithmic integrals associated to

Abstract

1 Introduction

2 Parameter differentiation of Mκ,μ\mathrm{M}_{\kappa,\mu}Mκ,μ​ via Kummer function 1F1{}_{1}F_{1}1​F1​

Definition 1

Theorem 2

2.1 Derivative with respect to the first parameter ∂Mκ,μ(x)/∂κ\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa∂Mκ,μ​(x)/∂κ

Theorem 3

Corollary 4

Lemma 5

Theorem 6

Corollary 7

2.2 Derivative with respect to the second parameter ∂Mκ,μ(x)/∂μ\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\mu∂Mκ,μ​(x)/∂μ

Theorem 8

Corollary 9

Theorem 10

3 Parameter differentiation of Mκ,μ\mathrm{M}_{\kappa,\mu}Mκ,μ​ via integral representations

3.1 Derivative with respect to the first parameter ∂Mκ,μ(x)/∂κ\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa∂Mκ,μ​(x)/∂κ

Definition 11

Definition 12

Theorem 13

Corollary 14

Theorem 15

Theorem 16

Theorem 17

Corollary 18

3.2 Application to the calculation of infinite integrals

Definition 19

Corollary 20

3.3 Derivative with respect to the second parameter ∂Mκ,μ(x)/∂μ\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\mu∂Mκ,μ​(x)/∂μ

Definition 21

Theorem 22

Theorem 23

Theorem 24

Corollary 25

3.4 Application to the calculation of finite integrals

Theorem 26

Theorem 27

4 Conclusions

Appendix A Parameter differentiation of the incomplete gamma functions

Definition 28

Definition 29

Theorem 30

Theorem 31

Corollary 32

Corollary 33

Corollary 34

Appendix B Reduction formulas for integral Whittaker functions Miκ,μ\mathrm{Mi}_{\kappa,\mu}Miκ,μ​ and miκ,μ\mathrm{mi}_{\kappa,\mu}miκ,μ​

Theorem 35

Remark 36

Theorem 37

Theorem 38

Remark 39

Appendix C Reduction formulas for the Whittaker function Mκ,μ(x)\mathrm{M}_{\kappa,\mu}\left(x\right)Mκ,μ​(x)

2 Parameter differentiation of $\mathrm{M}_{\kappa,\mu}$ via Kummer function ${}_{1}F_{1}$

2.1 Derivative with respect to the first parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa$

2.2 Derivative with respect to the second parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\mu$

3 Parameter differentiation of $\mathrm{M}_{\kappa,\mu}$ via integral representations

3.1 Derivative with respect to the first parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\kappa$

3.3 Derivative with respect to the second parameter $\partial\mathrm{M}_{\kappa,\mu}\left(x\right)/\partial\mu$

Appendix B Reduction formulas for integral Whittaker functions $\mathrm{Mi}_{\kappa,\mu}$ and $\mathrm{mi}_{\kappa,\mu}$

Appendix C Reduction formulas for the Whittaker function $\mathrm{M}_{\kappa,\mu}\left(x\right)$