A Quadruple Integral Containing the Gegenbauer Polynomial Cn(λ)(x): Derivation and Evaluation

Reynolds, Robert; Stauffer, Allan

doi:10.3390/sym14020205

Open AccessArticle

A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation

by

Robert Reynolds

^*

and

Allan Stauffer

Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

^*

Author to whom correspondence should be addressed.

Symmetry 2022, 14(2), 205; https://doi.org/10.3390/sym14020205

Submission received: 29 December 2021 / Revised: 19 January 2022 / Accepted: 19 January 2022 / Published: 21 January 2022

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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Abstract

:

A four-dimensional integral containing

g (x, y, z, t) C_{n}^{(λ)} (x)

is derived.

C_{n}^{(λ)} (x)

is the Gegenbauer polynomial,

g (x, y, z, t)

is a product of the generalized logarithm quotient functions and the integral is taken over the region

0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1, 0 \leq t \leq 1

. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. All the results in this work are new.

Keywords:

Gegenbauer polynomial; apéry’s constant; cauchy integral; quadruple integral

MSC:

Primary 30E20; 33-01; 33-03; 33-04; 33-33B; 33E20

1. Significance Statement

The definite integral of the Gegenbauer polynomial is evaluated in the work by Askey et al. [1]. In the work by Srivastava [2] the author obtained an inversion formula for a singular integral transform involving Gegenbauer polynomials. In the work done by Bingham [3] the author performed the passage to the limit so as to obtain a complete and explicit description of measures which is of importance in probabilistic work on random walks on spheres. The Gegenbauer polynomial has many mathematical applications which are detailed in Andrews et al. [4] (1999, Chapter 9). See also section (14.30) in [5]. Physical applications of these polynomials are detailed in section (18.38) in [5]. Other applications are detailed in section (18.39) in [5]. We extend the previous important work by adding three more dimensions to the previously derived integrals in this paper. A quadruple integral will be derived and expressed in terms of a Hurwitz–Lerch zeta function. The Hurwitz–Lerch zeta function

z e t a (s, v)

, the digamma function

p s i (0) (s)

, the Riemann zeta function

z e t a (k)

, and

l o g (2)

are used to deduce special cases.

2. Introduction

In this paper, we derive the quadruple definite integral given by

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} x^{m - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- m} (\frac{1}{t}) C_{n}^{(λ)} (x) {log}^{\frac{1}{2} (m - n - 1)} (\frac{1}{z}) \\ {log}^{λ + \frac{1}{2} (m + n - 1)} (\frac{1}{y}) {log}^{k} (\frac{a x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})}) d x d y d z d t \end{matrix}

(1)

where the parameters

k, a, λ, n

are general complex numbers and

1 / 2 < R e (m) < 1

. The method used by us in [6] are followed in the derivations. This method employs a form of the generalized Cauchy’s integral formula, which is given by

\frac{y^{k}}{Γ (k + 1)} = \frac{1}{2 π i} \int_{C} \frac{e^{w y}}{w^{k + 1}} d w .

(2)

where C is in general an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour.

3. Definite Integral of the Contour Integral

We use the method in [6,7]. The variable of integration in the contour integral is

r = w + m

. Using a generalization of Cauchy’s integral formula we form the quadruple integral by replacing y by

log (\frac{a x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})})

and multiplying by

x^{m - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- m} (\frac{1}{t}) C_{n}^{(λ)} (x) {log}^{\frac{1}{2} (m - n - 1)} (\frac{1}{z}) {log}^{λ + \frac{1}{2} (m + n - 1)} (\frac{1}{y})

then taking the definite integral with respect to

x \in [0, 1]

,

y \in [0, 1]

,

z \in [0, 1]

and

t \in [0, 1]

to obtain

\begin{matrix} \frac{1}{Γ (k + 1)} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} x^{m - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- m} (\frac{1}{t}) C_{n}^{(λ)} (x) {log}^{\frac{1}{2} (m - n - 1)} (\frac{1}{z}) \\ {log}^{λ + \frac{1}{2} (m + n - 1)} (\frac{1}{y}) {log}^{k} (\frac{a x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})}) d x d y d z d t \\ = \frac{1}{2 π i} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{C} a^{w} w^{- k - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} x^{m + w - 1} C_{n}^{(λ)} (x) {log}^{- m - w} (\frac{1}{t}) \\ {log}^{\frac{1}{2} (m - n + w - 1)} (\frac{1}{z}) {log}^{λ + \frac{1}{2} (m + n + w - 1)} (\frac{1}{y}) d w d x d y d z d t \\ = \frac{1}{2 π i} \int_{C} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} a^{w} w^{- k - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} x^{m + w - 1} C_{n}^{(λ)} (x) {log}^{- m - w} (\frac{1}{t}) \\ {log}^{\frac{1}{2} (m - n + w - 1)} (\frac{1}{z}) {log}^{λ + \frac{1}{2} (m + n + w - 1)} (\frac{1}{y}) d x d y d z d t d w \\ = \frac{1}{2 π i} \int_{C} \frac{π^{2} a^{w} w^{- k - 1} 2^{- 2 λ - m - w + 1} csc (π (m + w)) Γ (n + 2 λ)}{n! Γ (λ)} d w \end{matrix}

(3)

from Equation (18.17.37) in [5] and Equation (4.215.1) in [8] where

R e (m + w) > 0, R e (λ) > 0, R e (n) > 0, 1 / 2 < R e (m) < 1

and using the reflection Formula (8.334.3) in [8] for the Gamma function. The reversal of the order of integration over x, y, z and t is done by using Fubini’s theorem for multiple integrals see (9.112) in [9], since the integrand is of bounded measure over the space

C \times [0, 1] \times [0, 1] \times [0, 1] \times [0, 1]

.

4. The Hurwitz–Lerch Zeta Function and Infinite Sum of the Contour Integral

In this section, we use Equation (2) to derive the contour integral representations for the Hurwitz–Lerch Zeta function.

4.1. The Hurwitz–Lerch Zeta Function

The Hurwitz–Lerch zeta function see [5,10] has a series representation given by

Φ (z, s, v) = \sum_{n = 0}^{\infty} {(v + n)}^{- s} z^{n}

(4)

where

| z | < 1, v s . \neq 0, - 1, \dots

and is continued analytically by its integral representation given by

Φ (z, s, v) = \frac{1}{Γ (s)} \int_{0}^{\infty} \frac{t^{s - 1} e^{- v t}}{1 - z e^{- t}} d t = \frac{1}{Γ (s)} \int_{0}^{\infty} \frac{t^{s - 1} e^{- (v - 1) t}}{e^{t} - z} d t

(5)

where

R e (v) > 0

, and either

| z | \leq 1, z \neq 1, R e (s) > 0

, or

z = 1, R e (s) > 1

.

4.2. Infinite Sum of the Contour Integral

Using Equation (2) and replacing y by

log (a) + i π (2 y + 1) - log (2)

then multiplying both sides by

- \frac{i π^{2} 2^{- 2 λ - m + 2} e^{i π m (2 y + 1)} Γ (n + 2 λ)}{n! Γ (λ)}

taking the infinite sum over

y \in [0, \infty)

and simplifying in terms of the Hurwitz–Lerch zeta function we obtain

\begin{matrix} \frac{i^{k - 1} π^{k + 2} e^{i π m} 2^{k - 2 λ - m + 2} Γ (n + 2 λ) Φ (e^{2 i m π}, - k, \frac{- i log (a) + i log (2) + π}{2 π})}{k! n! Γ (λ)} \\ = - \frac{1}{2 π i} \sum_{y = 0}^{\infty} \int_{C} \frac{i π^{2} a^{w} w^{- k - 1} 2^{- 2 λ - m - w + 2} e^{i π (2 y + 1) (m + w)} Γ (n + 2 λ)}{n! Γ (λ)} d w \\ = - \frac{1}{2 π i} \int_{C} \sum_{y = 0}^{\infty} \frac{i π^{2} a^{w} w^{- k - 1} 2^{- 2 λ - m - w + 2} e^{i π (2 y + 1) (m + w)} Γ (n + 2 λ)}{n! Γ (λ)} \\ = \frac{1}{2 π i} \int_{C} \frac{π^{2} a^{w} w^{- k - 1} 2^{- 2 λ - m - w + 1} csc (π (m + w)) Γ (n + 2 λ)}{n! Γ (λ)} d w \end{matrix}

(6)

from Equation (1.232.2) in [8] where

I m (π (m + w)) > 0

in order for the sum to converge.

5. Definite Integral in Terms of the Hurwitz–Lerch Zeta Function

Theorem 1.

For all

k, a, λ, n \in C, 1 / 2 < R e (m) < 1

then,

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} x^{m - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- m} (\frac{1}{t}) C_{n}^{(λ)} (x) {log}^{\frac{1}{2} (m - n - 1)} (\frac{1}{z}) \\ {log}^{λ + \frac{1}{2} (m + n - 1)} (\frac{1}{y}) {log}^{k} (\frac{a x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})}) d x d y d z d t \\ = \frac{i^{k - 1} π^{k + 2} e^{i π m} 2^{k - 2 λ - m + 2} Γ (n + 2 λ) Φ (e^{2 i m π}, - k, \frac{- i log (a) + i log (2) + π}{2 π})}{n! Γ (λ)} \end{matrix}

(7)

Proof.

The right-hand sides of relations (3) and (6) are identical; hence, the left-hand sides of the same are identical too. Simplifying with the Gamma function yields the desired conclusion. □

Example 1.

The degenerate case

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} x^{m - 1} {(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- m} (\frac{1}{t}) C_{n}^{(λ)} (x) {log}^{\frac{1}{2} (m - n - 1)} (\frac{1}{z}) \\ {log}^{λ + \frac{1}{2} (m + n - 1)} (\frac{1}{y}) d x d y d z d t \\ = \frac{π^{2} 2^{- 2 λ - m + 1} csc (π m) Γ (n + 2 λ)}{n! Γ (λ)} \end{matrix}

(8)

Proof.

Use Equation (7) and set

k = 0

and simplify using entry (2) in Table below (64:12:7) in [11]. □

Example 2.

The Hurwitz zeta function

ζ (s, v)

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{{(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- \frac{n}{2} - \frac{1}{4}} (\frac{1}{z}) C_{n}^{(λ)} (x) {log}^{λ + \frac{n}{2} - \frac{1}{4}} (\frac{1}{y}) {log}^{k} (\frac{a x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})})}{\sqrt{x} \sqrt{log (\frac{1}{t})}} d x d y d z d t \\ = \frac{i^{k} π^{k + 2} 2^{2 k - 2 λ + \frac{3}{2}} Γ (n + 2 λ) ζ (- k, \frac{- i log (a) + i log (2) + π}{4 π})}{n! Γ (λ)} \\ - \frac{i^{k} π^{k + 2} 2^{2 k - 2 λ + \frac{3}{2}} Γ (n + 2 λ) ζ (- k, \frac{1}{2} (\frac{- i log (a) + i log (2) + π}{2 π} + 1))}{n! Γ (λ)} \end{matrix}

(9)

Proof.

Use Equation (7) and set

m = 1 / 2

and simplify using entry (4) in Table below (64:12:70) in [11]. □

Example 3.

The Digamma function

ψ^{(0)} (s)

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{{(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- \frac{n}{2} - \frac{1}{4}} (\frac{1}{z}) C_{n}^{(λ)} (x) {log}^{λ + \frac{n}{2} - \frac{1}{4}} (\frac{1}{y})}{\sqrt{x} \sqrt{log (\frac{1}{t})} log (\frac{a x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})})} d x d y d z d t \\ = \frac{i π 2^{- 2 λ - \frac{1}{2}} ψ^{(0)} (\frac{- i log (a) + i log (2) + π}{4 π}) Γ (n + 2 λ)}{n! Γ (λ)} \\ - \frac{i π 2^{- 2 λ - \frac{1}{2}} ψ^{(0)} (\frac{- i log (a) + i log (2) + 3 π}{4 π}) Γ (n + 2 λ)}{n! Γ (λ)} \end{matrix}

(10)

Proof.

Use Equation (9) and apply l’Hopital’s rule as

k \to - 1

and simplify using Equation (64:4:1) in [11]. □

Example 4.

The fundamental constant

log (2)

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{{(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- \frac{n}{2} - \frac{1}{4}} (\frac{1}{z}) C_{n}^{(λ)} (x) {log}^{λ + \frac{n}{2} - \frac{1}{4}} (\frac{1}{y})}{\sqrt{x} \sqrt{log (t)} (log (\frac{2 x \sqrt{log (y)} \sqrt{log (z)}}{log (t)}) + i π)} d x d y d z d t \\ = \frac{π 2^{\frac{1}{2} - 2 λ} log (2) Γ (n + 2 λ)}{n! Γ (λ)} \end{matrix}

(11)

Proof.

Use Equation (10) and set

a = - 2

and simplify. □

Example 5.

The Riemann zeta function

ζ (s)

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{{(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- \frac{n}{2} - \frac{1}{4}} (\frac{1}{z}) C_{n}^{(λ)} (x) {log}^{λ + \frac{n}{2} - \frac{1}{4}} (\frac{1}{y}) {log}^{k} (- \frac{2 x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})})}{\sqrt{x} \sqrt{log (\frac{1}{t})}} d x d y d z d t \\ = - \frac{i^{k} (2^{k + 1} - 1) π^{k + 2} 2^{k - 2 λ + \frac{3}{2}} ζ (- k) Γ (n + 2 λ)}{n! Γ (λ)} \end{matrix}

(12)

Proof.

Use Equation (7) and set

m = 1 / 2

and simplify using entry (4) in Table below (64:12:7) in [11]. □

Example 6.

Apéry’s constant

ζ (3)

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{{(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- \frac{n}{2} - \frac{1}{4}} (\frac{1}{z}) C_{n}^{(λ)} (x) {log}^{λ + \frac{n}{2} - \frac{1}{4}} (\frac{1}{y})}{\sqrt{x} \sqrt{log (\frac{1}{t})} {log}^{3} (- \frac{2 x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})})} d x d y d z d t \\ = \frac{3 i 2^{- 2 λ - \frac{7}{2}} ζ (3) Γ (n + 2 λ)}{π n! Γ (λ)} \end{matrix}

(13)

Proof.

Use Equation (12) and set

k = - 3

and simplify. □

Example 7.

The fundamental constant

ζ (5)

\begin{matrix} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{{(1 - x^{2})}^{λ - \frac{1}{2}} {log}^{- \frac{n}{2} - \frac{1}{4}} (\frac{1}{z}) C_{n}^{(λ)} (x) {log}^{λ + \frac{n}{2} - \frac{1}{4}} (\frac{1}{y})}{\sqrt{x} \sqrt{log (\frac{1}{t})} {log}^{5} (- \frac{2 x \sqrt{log (\frac{1}{y})} \sqrt{log (\frac{1}{z})}}{log (\frac{1}{t})})} d x d y d z d t \\ = - \frac{15 i 2^{- 2 λ - \frac{15}{2}} ζ (5) Γ (n + 2 λ)}{π^{3} n! Γ (λ)} \end{matrix}

(14)

Proof.

Use Equation (12) and set

k = - 5

and simplify. □

6. Discussion

This paper uses our contour integral method for deriving a new quadruple integral containing the Gegenbauer polynomial

C_{n}^{(λ)} (x)

, along with some interesting special cases with many more possible. The evaluations in this present work were numerically verified for both real and imaginary and complex values of the parameters in the integrals using Mathematica by Wolfram.

Author Contributions

Conceptualization, R.R.; methodology, R.R.; writing—original draft preparation, R.R.; writing—review and editing, R.R. and A.S.; funding acquisition, A.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research is supported by NSERC Canada under grant 504070.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Reynolds, R.; Stauffer, A. A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation. Symmetry 2022, 14, 205. https://doi.org/10.3390/sym14020205

AMA Style

Reynolds R, Stauffer A. A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation. Symmetry. 2022; 14(2):205. https://doi.org/10.3390/sym14020205

Chicago/Turabian Style

Reynolds, Robert, and Allan Stauffer. 2022. "A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation" Symmetry 14, no. 2: 205. https://doi.org/10.3390/sym14020205

APA Style

Reynolds, R., & Stauffer, A. (2022). A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation. Symmetry, 14(2), 205. https://doi.org/10.3390/sym14020205

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A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation

Abstract

1. Significance Statement

2. Introduction

3. Definite Integral of the Contour Integral

4. The Hurwitz–Lerch Zeta Function and Infinite Sum of the Contour Integral

4.1. The Hurwitz–Lerch Zeta Function

4.2. Infinite Sum of the Contour Integral

5. Definite Integral in Terms of the Hurwitz–Lerch Zeta Function

6. Discussion

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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