Part 2. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ function
Alexander Apelblat, Juan Luis Gonz\'alez-Santander
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 provides closed-form solutions for specific parameter values.
Contribution
It introduces new methods to compute parameter derivatives of the Whittaker function using infinite series and integrals, including closed-form expressions for special cases.
Findings
Derived derivatives of Whittaker function using hypergeometric functions
Expressed derivatives as infinite sums and integrals involving elementary functions
Provided closed-form solutions for particular parameter values
Abstract
First derivatives with respect to the parameters of the Whittaker function 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 infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of . These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function and its derivative with respect to , as well as some reduction formulas for the integral Whittaker…
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Taxonomy
TopicsMathematical functions and polynomials · Stochastic processes and financial applications
Part 2. Infinite series and logarithmic integrals associated to
differentiation with respect to parameters of the Whittaker 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 with respect to the parameters of the Whittaker function 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 infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of . These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function and its derivative with respect to , as well as some reduction formulas for the integral Whittaker functions and are calculated.
Keywords: Derivatives with respect to parameters; Whittaker functions; integral Whittaker functions; incomplete gamma functions; sums of infinite series of psi and gamma; infinite integrals involving Bessel functions.
AMS Subject Classification: 33B15, 33B20, 33C10, 33C15, 33C20, 33C50, 33E20.
1 Introduction
Two functions and were introduced to the mathematical literature by Whittaker [27] in 1903, and they are linearly independent solutions of the following second order differential equation:
[TABLE]
where and are parameters. For particular values of these parameters, the Whittaker functions and can be reduced to a variety of elementary and special functions (such as modified Bessel functions, incomplete gamma functions, parabolic cylinder functions, error functions, logarithmic and cosine integrals, as well as the generalized Hermite and Laguerre polynomials). Recently, Mainardi et al. [20] investigated the special case wherein the Wright function can be expressed in terms of Whittaker functions.
The Whittaker functions can be expressed as [22, Eqn. 13.14.2]:
[TABLE]
and [22, Eqn. 13.14.33]:
[TABLE]
where denotes the gamma function, and the Kummer function is defined as [21, Eqn. 47:3:1]:
[TABLE]
where denotes the Pochhammer polynomial and
[TABLE]
is the generalized hypergeometric function.
Also, the Whittaker function can be expressed as [22, Eqn. 13.14.3]:
[TABLE]
where denotes the Tricomi function.
Analytical properties of the Whittaker functions (see [11, 25, 28, 22, 19, 9, 14, 24, 21]) are of great interest in Mathematical Physics because these functions are involved in many applications, such as the solutions of the wave equation in paraboloidal coordinates, the behaviour of charges particles in fields with Coulomb potentials, stationary Green’s function in atomic and molecular calculations in Quantum Mechanics (i.e. solution of Schrödinger equation for the harmonic oscillator), probability density functions, and in many other physical and engineering problems [25, 17, 16, 23].
Mostly, the Whittaker functions are regarded as a function of variable with fixed values of parameters and , although there are few investigations where mathematical operations associated with both parameters are considered, especially for the parameter [17, 10, 1, 6]. In this context, it is worthwhile to mention Laurenzi’s paper [17], where the calculation of the derivative of with respect to when this parameter is an integrer is derived. In [10], Buschman showed that the derivative of with respect to the parameters can be expressed in terms of finite sums of these functions. Higher derivatives of the Whittaker functions with respect to parameter were discussed by Abad and Sesma [1], and integrals with respect to parameter by Becker [6]. Since the Whittaker functions are related to the confluent hypergeometric function, it is worth mention the investigation of the derivatives of the generalized hypergeometric functions presented by Ancarini and Gasaneo [2] or Sofostasios and Brychkov [26].
The integral Whittaker functions were introduced by us [4] as follows:
[TABLE]
In the current paper, the main attention will be devoted to Whittaker function by analyzing the first derivative of this function with respect to the parameters from the corresponding series and integral representations. Direct differentiation of the Whittaker functions leads to infinite sums of quotients of the digamma and gamma functions. It is possible to calculate these sums in closed-form in some cases with the aid of MATHEMATICA program. When the integral representations of the Whittaker function are taken into account, the results of differentiation can be expressed in terms of Laplace transforms of elementary functions. Integrands of the these Laplace type integrals include products of algebraic, exponential and logarithmic functions. New groups of infinite integrals are comparable to those investigated by Kölbig [15], Geddes et al. [12], and Apelblat and Kravitzky [5] are calculated in this paper.
Also, we will focus our attention on the integral Whittaker functions and in order to derive some new reduction formulas, as well as an integral representation of and its first derivative with respect to parameter .
2 Parameter differentiation of via Kummer function
Notation 1
Unless indicated otherwise, it is assumed throughout the paper that is a real variable and is a complex variable.
Definition 2
According to the notation introduced by Ancarini and Gasaneo [2], define
[TABLE]
and
[TABLE]
2.1 Derivative with respect to the first parameter
Taking into account (3) and (10), direct differentiation of (4) yields:
[TABLE]
If we apply first Kummer’s transformation formula [22, Eqn. 13.2.39]:
[TABLE]
we can rewrite (12) as
[TABLE]
Theorem 3
For , the following parameter derivative formula of holds true:
[TABLE]
where denotes the lower incomplete gamma function (248).
Proof. First note that
[TABLE]
since [22, Eqn. 13.14.31]:
[TABLE]
Now, let us calculate . For this purpose, take in (20) to obtain
[TABLE]
Note that according to [22, Eqn. 13.18.2]
[TABLE]
Also, from (3) and (19), we have
[TABLE]
Taking into account [21, Eqn. 45:6:2]:
[TABLE]
rewrite (44) as
[TABLE]
Consider as well the reduction formula given in the Appendix 254:
[TABLE]
Finally, according to the property [21, Eqn. 44:5:3]:
[TABLE]
see that
[TABLE]
Now, take the limit in (33), considering the results given in (31), (40), (45), (46) and (47), to obtain (27), as we wanted to prove.
Table 1 presents some explicit expressions for particular values of (27), obtained with the help of MATHEMATICA program.
Next, we present other reduction formula of from the result found in [17].
Theorem 4
The following reduction formula holds true for
[TABLE]
where denotes the Laguerre polynomial.
Proof. First note that, according to (31), we have
[TABLE]
Therefore, let us calculte . For this purpose, consider the formula [17]:
[TABLE]
Also, from [22, Eqn. 13.18.17], we have for
[TABLE]
thus applying (32) and taking in (51), we have
[TABLE]
Finally, insert (52) into (4) and consider (49) to obtain (4), as we wanted to prove.
In Table 2 we collect some particular cases of (4), obtained with the help of MATHEMATICA program.
Note that for , we obtain an indeterminate expression in (4). We calculate this particular case with a result of the next Section.
Theorem 5
The following reduction formula holds true:
[TABLE]
where G_{p,q}^{m,n}\left(z\left|\begin{array}[]{c}a_{1},\ldots,a_{p}\\ b_{1},\ldots,b_{q}\end{array}\right.\right) denotes the Meijer-G function.
Proof. According to [22, Eqn. 13.18.2], we have
[TABLE]
thus, performing the derivative with respect to ,
[TABLE]
Taking and considering (49), we have
[TABLE]
Finally, apply (65) and (67), to arrive at (53) as we wanted to prove.
2.2 Derivative with respect to the second parameter
Theorem 6
For , the following parameter derivative formula of holds true:
[TABLE]
Proof. Differentiate the following reduction formula with respect to parameter [22, Eqn. 13.18.2]:
[TABLE]
to obtain
[TABLE]
Insert (27) in (64) to arrive at (60), as we wanted to prove.
Table 3 shows the derivative of with respect for particular values of and using (60) and the help of MATHEMATICA program.
Theorem 7
The following parameter derivative formula of holds true:
[TABLE]
where denotes the modified Bessel of the second kind (Macdonald function).
Proof. Differentiate with respect to the expression [22, Eqn. 13.18.9]:
[TABLE]
to obtain
[TABLE]
as we wanted to prove.
The order derivative of is given in terms of Meijer-G functions for, and [13]:
[TABLE]
where is the modified Bessel function; or in terms of generalized hypergeometric functions for, , and [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 (65), (67) and (73), some derivatives of with respect has been calculated with the help of MATHEMATICA program, and they are presented in Table 4.
3 Parameter differentiation of via integral representations
3.1 Derivative with respect to the first parameter
Integral representations of the Whittaker function for and are given in the form of Laplace transform [19, Sect. 7.4.2]:
[TABLE]
and as the infinite integral:
[TABLE]
In order to calculate the first derivative of with respect to parameter , let us introduce the following finite logarithmic integrals.
Definition 8
For and , define:
[TABLE]
For , differentiation of (3.1) and (3.1) with respect to parameter yields respectively
[TABLE]
Note that, from (90) and (91), we have
[TABLE]
Theorem 9
The following integral holds true for and :
[TABLE]
where denotes the beta function.
Proof. Compare (12) to (90) and take into account (3) to arrive at (101), as we wanted to prove.
Now, we derive a Lemma that will be applied throughout this Section and the next one.
Lemma 10
For and , the following Laplace transform holds true:
[TABLE]
where and denotes respectively the upper and lower incomplete gamma functions, (248) and (250).
Proof. Split the integral in two terms as follows:
[TABLE]
and apply the Laplace transform for [24, Eqn. 2.5.2(4)]111It is worth noting that there is an incorrect sign in the reference cited.:
[TABLE]
to obtain
[TABLE]
and
[TABLE]
Note that, according to Kummer’s transformation (19), and to the reduction formula [24, Eqn. 7.11.1(14)]:
[TABLE]
we have for
[TABLE]
thus (108) becomes
[TABLE]
Now, insert (115) and (124) in (187) to arrive at
[TABLE]
Next, apply the transformation formula [24, Eqn. 7.12.1(7)]:
[TABLE]
taking and , and applying again (3.1), to arrive at
[TABLE]
[TABLE]
Applying the properties [21, Eqn. 45:0:1]
[TABLE]
and [18, Eqn. 1.2.2]
[TABLE]
rewrite (140) as (10), as we wanted to prove.
Theorem 11
The following integral holds true for and :
[TABLE]
Proof. From (88) and (10), we obtain the desired result.
Remark 12
If we insert (125) in (143), we obtain the following alternative form:
[TABLE]
Theorem 13
The following reduction formula holds true for and :
[TABLE]
Proof. Insert in (90) the reduction formula [22, Eqn. 13.18.2] with , i.e.
[TABLE]
and the result given in (142) to arrive at (151).
Remark 14
If we consider (147), we obtain the following alternative form:
[TABLE]
Table 5 shows the first derivative of with respect to parameter for some particular values of and , and , calculated with the aid of MATHEMATICA program from (156).
Notice that for , we obtain an indeterminate expression in (151) and (156). For these cases, we present the following result.
Theorem 15
The following reduction formula holds true for :
[TABLE]
Proof. Take in (156) and perform the limit
[TABLE]
On the one hand, let us prove the following asymptotic formulas for
[TABLE]
In order to prove (165), consider [21, Eqn. 44:5:4]
[TABLE]
thus, knowing that [18, Eqn. 1.3.6]
[TABLE]
and performing the substitution , we have
[TABLE]
where denotes the -th harmonic number. In order to prove (166), note that and for we have the expansion [21, Eqn. 44:6:2]
[TABLE]
Finally, notice that (167) follows directly from [18, Eqn. 1.1.5]. Therefore, taking into account (165)-(167), and taking into account (141), we conclude
[TABLE]
Insert (3.1) in (161) to arrive at
[TABLE]
On the other hand, consider the reduction formula (259), derived in the Appendix,
[TABLE]
and the formula [22, Eqn. 8.4.15]
[TABLE]
where denotes the exponential integral [22, Eqn. 6.2.1], which is defined as
[TABLE]
where the path does not cross the negative real axis or pass throught the origin. Also, consider the property [22, Eqn. 6.2.4]
[TABLE]
Therefore, substituting (174) and (175) in (170), and taking into account (176), we arrive at (160), as we wanted to prove.
Remark 16
It is worth noting that from [10],
[TABLE]
where and are integers of like parity, we can derive an equivalent reduction formula to (160). Indeed, taking , (177) is reduced to
[TABLE]
Note that from (59), we have
[TABLE]
Also, from (7) and the reduction formula for given in [22, Eqn. 13.2.8]
[TABLE]
we obtain
[TABLE]
Therefore, susbtituting (179) and (180) in (178), and simplifying, we arrive at
[TABLE]
Perform the index substitution and exchange the sum order in (181), to arrive at
[TABLE]
By virtue of the binomial theorem, the inner sum in (182) is just , thus we finally obtain:
[TABLE]
Theorem 17
For , and , the following integral holds true:
[TABLE]
Proof. From (88), we have
[TABLE]
thus, taking with and applying the binomial theorem, we get
[TABLE]
Insert the result obtained in (10) for in (185) to arrive at
[TABLE]
Now, take into account (174), to get
[TABLE]
Finally, note that using the exponential polynomial, defined as
[TABLE]
and the property for [21, Eqn. 45:4:2]:
[TABLE]
we calculate the following finite sum as:
[TABLE]
Apply (188) to (187) in order to obtain (184), as we wanted to prove.
Theorem 18
For , the following reduction formula holds true:
[TABLE]
Proof. Applying (7) and [22, Eqn. 13.6.6]
[TABLE]
see that for
[TABLE]
Taking into account (168) and (176), insert (184) and (190) in (90) for and to arrive at (18), as we wanted to prove.
Theorem 19
For , and , the following integral holds true:
[TABLE]
Proof. Applying the binomial theorem to (88) for and , we have
[TABLE]
Insert the result obtained in (10) for in (192) to get
[TABLE]
Now, take into account (174), to obtain
[TABLE]
Finally, consider [22, Eqns. 10.47.9,12]
[TABLE]
where is the modified spherical Bessel function of the second kind, to arrive at the desired result.
Theorem 20
For , the following reduction formula holds true:
[TABLE]
Proof. Take and in (90), to obtain
[TABLE]
Consider [18, Eqn. 1.3.7]
[TABLE]
and [22, Eqns. 13.18.9]
[TABLE]
Substitute (191), (197) and (198) in (196), and take into account (31) and (176), to arrive at (20), as we wanted to prove.
Table 6 shows the first derivative of with respect to parameter for some particular values of and , calculated with the aid of MATHEMATICA from (160), (18) and (20).
3.2 Application to the calculation of infinite integrals
Additional integral representations of the Whittaker function in terms of Bessel functions [19, Sect. 7.4.2] are known:
[TABLE]
Let us introduce the following infinite logarithmic integral.
Definition 21
[TABLE]
Theorem 22
For with , the following integral holds true:
[TABLE]
where is given by (101).
Proof. Differentiation of (3.2) with respect to parameter yields:
[TABLE]
Equate (90) to (202) to arrive at (201), as we wanted to prove.
3.3 Derivative with respect to the second parameter
First, note that
[TABLE]
since (32) is satisfied. Next, let us introduce the following definitions in order to calculate the first derivative of with respect to parameter .
Definition 23
Following the notation introduced in (10)-(11), define
[TABLE]
and
[TABLE]
Direct differentiation of (7) yields:
[TABLE]
Definition 24
For and , define:
[TABLE]
These integrals are interrelated by
[TABLE]
Differentiation of (3.1) with respect to parameter gives
[TABLE]
Theorem 25
According to the notation introduced in (204) and (205), the following integral holds true for :
[TABLE]
Proof. Comparing (3.3) to (3.3), taking into account (7), we arrive at (25), as we wanted to prove.
Theorem 26
For and , the following reduction formula holds true:
[TABLE]
Proof. According to (10) and (207), note that
[TABLE]
Taking in (3.3), substitute (215) and (155) to arrive at the desired result given in (211).
Remark 27
If we take into account (125) in (215), we obtain the alternative form:
[TABLE]
thus for and , we have
[TABLE]
Table 7 shows the first derivative of with respect to parameter for some particular values of and , with , calculated from (221) with the aid of MATHEMATICA program.
Notice that for , we obtain an indeterminate expression in (211) or (221). For these cases, we present the following result.
Theorem 28
The following reduction formula holds true for :
[TABLE]
Proof. Take in (221) and perform the limit
[TABLE]
Applying the result given in (3.1), we get
[TABLE]
Now, compare (160) to (170), to see that
[TABLE]
Therefore, inserting (234) in (227), and taking into account (203), we arrive at (28), as we wanted to prove.
Remark 29
It is worth noting that from [10],
[TABLE]
where and are integers of like parity, we can derive an equivalent reduction formula to (28). Indeed, following similar steps as in Remark 16, we arrive at:
[TABLE]
Theorem 30
For , the following reduction formula holds true:
[TABLE]
Proof. According to (207) and (10), using the binomial theorem, and taking into account (197), we have
[TABLE]
Consider (174), (176) and (188) in order to rewrite (241) as
[TABLE]
Therefore, substituting (190), (168), and (242) in (3.3), we obtain (30), as we wanted to prove.
Theorem 31
For , the following reduction formula holds true:
[TABLE]
Proof. Applying the binomial theorem to (207) for and , and taking into account (10), (174), (176), and (194) for , we arrive at
[TABLE]
Take and in (3.3), and substitute (3.3) and (198) in order to arrive at (31), as we wanted to prove.
Table 8 shows with respect to parameter for some particular values of and , which has been calculated from (28), (30), and (31) with the aid of MATHEMATICA program.
4 Integral Whittaker functions and
In [4], we found some reduction formulas for the integral Whittaker function . Next, we derive some new reduction formulas for and from reduction formulas of the Whittaker function .
Theorem 32
The following reduction formula holds true for and :
[TABLE]
Proof. According to [22, Eqn. 13.18.17]
[TABLE]
where [18, Eqn. 4.17.2]
[TABLE]
denotes the Laguerre polynomials. Insert (247) in (246) and integrate term by term according to the definition of the integral Whittaker function (8), to get
[TABLE]
Finally, take into account the defintion of the lower incomplete gamma function [22, Eqn. 8.2.1]:
[TABLE]
and simplify the result to arrive at (245), as we wanted to prove.
Remark 33
Taking in (245), we recover the formula given in [4].
Theorem 34
The following reduction formula holds true for , and :
[TABLE]
where denotes the upper incomplete gamma function (250).
Proof. Follow similar steps as in the previous theorem, but consider the definition of the upper incomplete gamma function [22, Eqn. 8.2.2]:
[TABLE]
Theorem 35
The following reduction formula holds true for , and :
[TABLE]
Proof. From (194) and (198), we have
[TABLE]
thus, integrating term by term, we obtain
[TABLE]
Finally, taking into account (250), we arrive at (251), as we wanted to prove.
Theorem 36
For and , the following integral representation holds true:
[TABLE]
Proof. According to (9) and (3.1), we have
[TABLE]
Exchange the integration order and calculate the inner integral using (250), to arrive at (252), as we wanted to prove.
Remark 37
It is worth noting that we cannot follow the above steps to derive the integral representation of because the corresponding integral does not converge, except for some special cases such as the ones given in (245).
Theorem 38
For and , the following integral representation holds true:
[TABLE]
Proof. Direct differentiation of (252) with respect to yields (253), as we wanted to prove.
5 Conclusions
The Whittaker function is defined in terms of the Tricomi function, hence its derivative with respect to the parameters and can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, the parameter differentiation of some integral representations of leads to infinite integrals of elementary functions. These sums and integrals has been calculated for some particular cases of the parameters and in closed-form. As an application of these results, we have calculated an infinite integral containing the Macdonald function. 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 calculate a reduction formula for the first derivative of the Kummer function, i.e. , which it is necessary for the derivation of Theorem 3.
In the second Appendix, we calculate a reduction formula of the hypergeometric function for non-negative integer , since it is not found in most common literature, such as [24]. This reduction formula is used throughout Section 3 in order to simplify the results obtained.
Finally, we collect some reduction formulas for the Whittaker function in the last Appendix.
Appendix A Calculation of
Theorem 39
The following reduction formula holds true:
[TABLE]
Proof. According to the definition of the Kummer function (5), we have
[TABLE]
Taking into account [21, Eqn. 18:5:7]
[TABLE]
and the definition of the generalized hypergeometric function (6), we may recast (255) as
[TABLE]
thus, for , we obtain222It is worth noting that there is a typo in [24, Eqn. 7.12.1(5)].
[TABLE]
Applying L’Hôpital’s rule, calculate the limit in (256), considering the notation given in (10),
[TABLE]
Finally, differentiate Kummer’s transformation formula (19) with respect to the first parameter to obtain:
[TABLE]
Apply (258) in order to rewrite (257) as (254), as we wanted to prove.
Appendix B Calculation of
Theorem 40
For , the following reduction formula holds true:
[TABLE]
where denotes the complementary exponential integral.
Proof. Consider the function
[TABLE]
thus
[TABLE]
and by induction
[TABLE]
Now, apply the repeated integral formula [22, Eqn. 1.4.31]
[TABLE]
to obtain
[TABLE]
Use the binomial theorem to expand (263) as
[TABLE]
According to [22, Eqn. 6.2.3], we have
[TABLE]
Also, taking into account the definition of the lower incomplete gamma function [21, Eqn. 45:3:1], we calculate for
[TABLE]
Therefore, substituting (267) and (268) in (B), we have
[TABLE]
Finally, consider the formula [14, Eqn. 0.155.4]
[TABLE]
to arrive at (259), as we wanted to prove
Appendix C Reduction formulas for the Whittaker function
For convenience of the readers, reduction formulas for the Whittaker function are presented in their explicit form in Table 9.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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