A Note About the {Ki(z)} Functions

TL;DR

This paper simplifies proofs of properties of the $K_{i}(z)$ functions, addresses an open question from prior work, and explores their differential transcendency, contributing to the theoretical understanding of these special functions.

Contribution

It provides simplified proofs of known properties, resolves an open question, and discusses the differential transcendency of the $K_{i}(z)$ functions.

Findings

Simplified proofs of two properties of $K_{i}(z)$ functions.

Resolution of an open question from previous research.

Discussion on the differential transcendency of $K_{i}(z)$ functions.

Abstract

In the article [Petojevic 2006], A. Petojevi\' c verified useful properties of the $K_{i}(z)$ functions which generalize Kurepa's [Kurepa 1971] left factorial function. In this note, we present simplified proofs of two of these results and we answer the open question stated in [Petojevic 2006]. Finally, we discuss the differential transcendency of the $K_{i}(z)$ functions.

Equations

$${}_{v}M_{m}(s;a,z)=\displaystyle\sum\limits_{k=1}^{v}{(-1)^{k-1}{z+m+1-k\choose m+1}\mbox{$\cal L$}[s;{}_{2}F_{1}(a,k-z,m+2;1-t)]},$$

$$K(z)=\underset{0}{\overset{\infty }{\int }}{e}^{-t}\frac{t^z-1}{t-1}dt,$$

$$K_i(z)=\frac{{}_1{M}_0(1;1,z+i-1)-{}_1{M}_0(1;1,i-1)}{{}_1{M}_{-1}(1;1,i)},$$

$$K_{i}(z)=\displaystyle\frac{1}{(i\!-\!1)!}{\Big{(}}K(z+i-1)-K(i-1){\Big{)}},$$

$$\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-1$}}{K(z)}=-1\;\quad\;\mbox{\rm and}\quad\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-m$}}{K(z)}=\displaystyle\sum\limits_{k=2}^{m-1}{\displaystyle\frac{(-1)^{k-1}}{k!}},\;\;(m\!\geq\!3).$$

$$\;\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-(i\!+\!m)$}}{\!\!\!\!K_{i}(z)}=\displaystyle\frac{1}{(i\!-\!1)!}\;\;\cdot\!\!\!\!\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-(i\!+\!m)$}}{\!\!\!\!K(z+i-1)}=\displaystyle\frac{1}{(i\!-\!1)!}\;\;\cdot\!\!\!\!\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-(m\!+\!1)$}}{\!\!K(z)},$$

$$\;\;\;\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-i$}}{\!\!K_{i}(z)}\!=\!-\displaystyle\frac{1}{(i\!-\!1)!}\;\;\quad\mbox{\rm and}\quad\!\!\!\!\mathop{\mbox{\rm res}}\limits_{\mbox{\tiny$z=-(i\!+\!m)$}}{\!\!K_{i}(z)}\!=\!\displaystyle\frac{1}{(i\!-\!1)!}\cdot\!\!\displaystyle\sum_{k=2}^{m}{\displaystyle\frac{(-1)^{k-1}}{k!}},\;\;(m\!\geq\!2).$$

$$\displaystyle\frac{K_{i}(x)}{\Gamma(x+i-1)}=\displaystyle\frac{1}{(i\!-\!1)!}\cdot\displaystyle\frac{K(i+x-1)-K(i-1)}{\Gamma(x+i-1)}\mathop{\longrightarrow}\limits_{\mbox{\tiny$x\!\rightarrow\!\infty$}}\displaystyle\frac{1}{(i\!-\!1)!}$$

$$\displaystyle\frac{K_{i}(x)}{\Gamma(x+i)}=\displaystyle\frac{1}{(i\!-\!1)!}\cdot\displaystyle\frac{K(i+x-1)-K(i-1)}{(x+i-1)\Gamma(x+i-1)}\mathop{\longrightarrow}\limits_{\mbox{\tiny$x\!\rightarrow\!\infty$}}0.$$

$$K(z)=\frac{\text{mbox}\mathrm{Ei}(1)+\text{mbox}\text{neweu}i\pi }{e}+\frac{(-1{)}^z\Gamma (1+z)\Gamma (-z,-1)}{e},$$

$$\begin{array}[]{rcl}K_{i}(z)&\!\!=\!\!&\displaystyle\frac{1}{(i\!-\!1)!}{\Big{(}}K(z+i-1)-K(i-1){\Big{)}}\\[8.61108pt] &\!\!=\!\!&\displaystyle\frac{\mbox{\rm Ei}(1)+\mbox{\neweu i}\,\pi}{e(i\!-\!1)!}+\displaystyle\frac{(-1)^{z+i-1}\Gamma(1+z+i-1)\Gamma(-z-i+1,-1)}{e(i\!-\!1)!}\\[8.61108pt] &&\!\!\!\!-\displaystyle\frac{\mbox{\rm Ei}(1)+\mbox{\neweu i}\,\pi}{e(i\!-\!1)!}-\displaystyle\frac{(-1)^{i-1}\Gamma(i)\Gamma(-i+1,-1)}{e(i\!-\!1)!}\\[7.5347pt] &\!\!=\!\!&(-1)^{i}e^{-1}{\bigg{(}}\Gamma(1-i,-1)-(-1)^{z}\displaystyle\frac{\Gamma(1-i-z,-1)\Gamma(i+z)}{(i\!-\!1)!}{\bigg{)}}.\end{array}$$

Full text

A NOTE ABOUT THE $\{K_{i}(z)\}_{i=1}^{\infty}$ FUNCTIONS

††footnotetext: Research partially supported by the MNTRS, Serbia, Grant No. 144020.

Branko J. Malešević

In the article [11], A. Petojević verified useful properties of the $K_{i}(z)$ functions which generalize Kurepa’s [2] left factorial function. In this note, we present simplified proofs of two of these results and we answer the open question stated in [11]. Finally, we discuss the differential transcendency of the $K_{i}(z)$ functions.

A. Petojević [8, p. 3.] considered the family of functions:

[TABLE]

for $\Re(z)\,\!>\!\,v\!-\!m\!-\!2$, where $v\!\in\!\mbox{\bf N}$ is a positive integer; $m\!\in\!\{-1,0,1,2,\ldots\}$ is an integer; $s,a,z$ are complex variables; \mbox{\cal L}[s;F(t)] is Laplace transform and ${}_{2}F_{1}(a,b,c;x)$ is the hypergeometric function $(|x|\!<\!1)$. D-  . Kurepa has considered in the articles [2, p. 151.] and [3, p. 297.] a complex function defined by the integral:

[TABLE]

for $\Re(z)\!>\!0$. Especially, for Kurepa’s function $K(z)$, it is true that $K(z)\!=\!{}_{1}M_{0}(1;1,z)$, for $\Re(z)\!>\!0$, according to [11]. For various of values of parameters $v,m,s,a,z$ from (1), different special functions, as presented in [11], are obtained. A. Petojević has considered in the article [11, p. 1640.] the following sequence of functions:

[TABLE]

for $i\!\in\!\mbox{\bf N}$ and $\Re(z)\!>\!-i$. On the basis of the definition in (3), the following representation via Kurepa’s function is true:

[TABLE]

for $i\!\in\!\mbox{\bf N}$ and $\Re(z)\!>\!-i+1$. Note that $K(0)\!=\!0$ [3, p. 297.] and therefore $K_{1}(z)\!=\!K(z)$ for $\Re(z)\!>\!0$. Analytical and differential–algebraic properties of Kurepa’s function $K(z)$ are considered in articles $[1-12]$ and in many other articles. On the basis of well-known statements for Kurepa’s function $K(z)$, using representation $(\ref{Eq_4})$, in many cases we can get simple proofs for analogous statements for $K_{i}(z)$ functions. For example, it is a well-known fact that it is possible to analytically continue Kurepa’s function to a meromorphic function with simple poles at integer points $z=-1$ and $z=-m$, ($m\geq 3$) [3, p. 303.], [4, p. 474.]. Residues of Kurepa’s function at these poles have the following form [3]:

[TABLE]

For Kurepa’s function $K(z)$ the infinite point is an essential singularity [4]. Hence, on the basis of (4), each function $K_{i}(z)$ is meromorphic with simple poles at integer points $z\!=\!-i\,$ and $\,z\!=\!-(i+m)$, $(m\!\geq\!2)$. On the basis of (4) we have:

[TABLE]

where $m=0$ or $m\!\geq\!2$. Hence:

[TABLE]

For each $K_{i}(z)$ function the infinite point is an essential singularity. Therefore, we get Theorem 3.3. from [11]. Next, it is a well-known fact that for Kurepa’s function the following asymptotic relation $K(x)\sim\Gamma(x)$ is true for real $x$ such that $x\rightarrow\infty$ and where $\Gamma(x)$ is the gamma function [3, p. 299.]. Hence, for fixed $i\!\in\!\mbox{\bf N}$ and real $x\!>\!-i\!+\!1$, on the basis of (4), we get:

[TABLE]

and

[TABLE]

Therefore, we get Theorem 3.6. from [11]. Next we give a solution to the open problem stated in Question 3.7. in [11]. Namely, the following formula in the article [9, p. 35.] is given:

[TABLE]

for values $z\!\in\!\mbox{\bf C}\backslash\{-1,-2,-3,-4,\ldots\}$ and $\mbox{\neweu i}\!=\!\sqrt{-1}$. In the previous formula $\mbox{\rm Ei}(z)$ and $\Gamma(z,a)$ are exponential integral and incomplete gamma function respectively [9]. Then, for fixed $i\!\in\!\mbox{\bf N}$ and values $z\!\in\!\mbox{\bf C}\backslash\{-i,-i-1,-i-2,-i-3,\ldots\}$, on the basis of (4) and (10), we get:

[TABLE]

Therefore, the affirmative answer for Question 3.7. from [11] is true for complex values $z\!\in\!\mbox{\bf C}\backslash\{-i,-i-1,-i-2,-i-3,\ldots\}$.

Finally, at the end of this note let us emphasize one differential–algebraic fact for the sequence of functions $K_{i}(z)$. On the basis of the formula (17) from the article [11], we can conclude that each $K_{i}(z)$ function satisfies the following recurrence relation $(i\!-\!1)!\,K_{i}(z+1)-(i\!-\!1)!\,K_{i}(z)=\Gamma(z+i)$. The previous relation can be used to verify the differential transcendency of these functions as discussed in [12, 13]. Therefore, we can conclude that each $K_{i}(z)$ function is a differential transcendental function, i.e. it satisfies no algebraic differential equation over the field of complex rational functions.