EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1113-1119 ISSN 1307-5543 – ejpam.com Published by New York Business Global A quadruple integral involving the Legendre function Pn(x) of the first kind: derivation and evaluation Robert Reynolds1,∗, Allan Stauffer1 1 Department of Mathematics and Statistics, Faculty of Science, York University, Toronto, Ontario, Canada, M3J1P3 Abstract. A closed form expression of a quadruple integral involving the Legerndre polynomial Pn(x) is derived. Special cases are expressed in terms of special functions and fundamental con- stants. All the results in this work are new. 2020 Mathematics Subject Classifications: 30E20, 33-01, 33-03, 33-04, 33-33B Key Words and Phrases: Legendre polynomial, quadruple integral, Hurwitz-Lerch zeta func- tion, Cauchy integral formula 1. Significance Statement The Legendre functions are the most well known particular cases of the hypergeometric function. They have been discovered by Laplace and Legendre as early as in the 18th century. Later on their importance has grown substantially due to their connections with many problems of mathematical physics [3]. In this present work we investigate the quadruple integral involving the Legendre polynomial Pn(x) and the invariance of the parameter n with respect to the Hurwitz-Lerch zeta function. 2. Introduction In this paper we derive the quadruple definite integral given by∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 xm−1Pv(x) log −m ( 1 t ) log 1 2 (m−v−1) ( 1 y ) log m+v 2 ( 1 z ) logk ax √ log ( 1 y )√ log ( 1 z ) log ( 1 t )  dxdydzdt (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4237 Email addresses: milver@my.yorku.ca (R. Reynolds), stauffer@yorku.ca (A. Stauffer) https://www.ejpam.com 1113 © 2022 EJPAM All rights reserved. R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 1113-1119 1114 where the parameters k, a, v,m are general complex numbers and Re(v) < Re(m) < 1/2. This definite integral will be used to derive special cases in terms of special functions and fundamental constants. The derivations follow the method used by us in [6]. This method involves using a form of the generalized Cauchy’s integral formula given by yk Γ(k + 1) = 1 2πi ∫ C ewy wk+1 dw. (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. We then multiply both sides by a function of x, y, z and t, then take a definite quadruple integral of both sides. This yields a definite integral in terms of a contour integral. Then we multiply both sides of Equation (2) by another function of x, y, z and t and take the infinite sums of both sides such that the contour integral of both equations are the same. 3. Definite Integral of the Contour Integral We use the method in [6]. The variable of integration in the contour integral is u = w + m. The cut and contour are in the first quadrant of the complex u-plane. The cut approaches the origin from the interior of the first quadrant and the contour goes round the origin with zero radius and is on opposite sides of the cut. Using a generalization of Cauchy’s integral formula we form the triple integral by replacing y by log ax √ log ( 1 y )√ log( 1 z ) log( 1 t )  and multiplying by xm−1Pv(x) log −m ( 1 t ) log 1 2 (m−v−1) ( 1 y ) log m+v 2 ( 1 z ) then taking the definite integral with respect to x ∈ [0, 1], y ∈ 0, 1], z ∈ 0, 1] and t ∈ 0, 1] to obtain 1 Γ(k + 1) ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 xm−1Pv(x) log −m ( 1 t ) log 1 2 (m−v−1) ( 1 y ) log m+v 2 ( 1 z ) logk ax √ log ( 1 y )√ log ( 1 z ) log ( 1 t )  dxdydzdt = 1 2πi ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ C aww−k−1Pv(x)x m+w−1 log−m−w ( 1 t ) log 1 2 (m−v+w−1) ( 1 y ) log 1 2 (m+v+w) ( 1 z ) dwdxdydzdt = 1 2πi ∫ C ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 aww−k−1Pv(x)x m+w−1 log−m−w ( 1 t ) R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 1113-1119 1115 log 1 2 (m−v+w−1) ( 1 y ) log 1 2 (m+v+w) ( 1 z ) dxdydzdtdw = 1 2πi ∫ C π3/2aww−k−12−m−w csc(π(m+ w))dw (3) from equation (1.8.8.1) in [4] and equation (4.215.1) in [2] where Re(π(m + w)) > 0 and using the reflection formula (8.334.3) in [2] for the Gamma function. We are able to switch the order of integration over x, y, z and t using Fubini’s theorem since the integrand is of bounded measure over the space C× [0, 1]× [0, 1]× [0, 1]× [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 (25.14) in [1] has a series representation given by Φ(z, s, v) = ∞∑ n=0 (v + n)−szn (4) where |z|< 1, v ̸= 0,−1, .. and is continued analytically by its integral representation given by Φ(z, s, v) = 1 Γ(s) ∫ ∞ 0 ts−1e−vt 1− ze−t dt = 1 Γ(s) ∫ ∞ 0 ts−1e−(v−1)t et − z dt (5) where Re(v) > 0, and either |z|≤ 1, z ̸= 1, Re(s) > 0, or z = 1, Re(s) > 1. 4.2. Infinite sum of the Contour Integral Using equation (2) and replacing y by log(a) + iπ(2y + 1) − log(2) then multiplying both sides by −iπ3/221−meiπm(2y+1) taking the infinite sum over y ∈ [0,∞) and simplifying in terms of R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 1113-1119 1116 the Hurwitz-Lerch zeta function we obtain (6) 1 Γ(k + 1) ik−1πk+ 3 2 eiπm2k−m+1Φ ( e2imπ,−k, −i log(a) + i log(2) + π 2π ) = − 1 2πi ∞∑ y=0 ∫ C iπ3/2aww−k−12−m−w+1eiπ(2y+1)(m+w)dw = − 1 2πi ∫ C ∞∑ y=0 iπ3/2aww−k−12−m−w+1eiπ(2y+1)(m+w)dw = 1 2πi ∫ C π3/2aww−k−12−m−w csc(π(m+ w))dw from equation (1.232.3) in [2] where Im(π(m+ w)) > 0 in order for the sum to converge. 5. Definite Integral in terms of the Lerch Function and invariant index forms Theorem 1. For all k, a, v,m ∈ C, Re(v) < Re(m) ≤ 1/2,∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 xm−1Pv(x) log −m ( 1 t ) log 1 2 (m−v−1) ( 1 y ) log m+v 2 ( 1 z ) logk ax √ log ( 1 y )√ log ( 1 z ) log ( 1 t )  dxdydzdt = ik−1πk+ 3 2 eiπm2k−m+1Φ ( e2imπ,−k, −i log(a) + i log(2) + π 2π ) (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.∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 xm−1Pv(x) log −m ( 1 t ) log 1 2 (m−v−1) ( 1 y ) log m+v 2 ( 1 z ) dxdydzdt = π3/22−m csc(πm) (8) Proof. Use equation (7) and set k = 0 and simplify using entry (2) in Table below (64:12:7) in [5]. R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 1113-1119 1117 Example 2. The Hurwitz zeta function ζ(k, a), where the right-hand side is invariant with respect to v, (9 ) ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 Pv(x) log 1 2(−v− 1 2) ( 1 y ) log 1 2(v+ 1 2) ( 1 z ) √ x √ log ( 1 t ) logk ax √ log ( 1 y )√ log ( 1 z ) log ( 1 t )  dxdydzdt = ik2k+ 1 2πk+ 3 2 ( 2kζ ( −k, −i log(a) + i log(2) + π 4π ) − 2kζ ( −k, 1 2 ( −i log(a) + i log(2) + π 2π + 1 ))) Proof. Use equation (7) and set m = 1/2 and simplify using entry (4) in Table below (64:12:7) in [5]. Example 3. The zeta function of Riemann ζ(k), ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 Pv(x) log − v 2 − 1 4 ( 1 y ) log v 2 + 1 4 ( 1 z ) √ x √ log ( 1 t ) logk − 2x √ log ( 1 y )√ log ( 1 z ) log ( 1 t )  dxdydzdt = −ik2k+ 1 2 ( 2k+1 − 1 ) πk+ 3 2 ζ(−k) (10) Proof. Use equation 9 and set a = −2 and simplify using entry (2) in Table below (64:7) in [5]. Example 4. The fundamental constant log(2), (11 ) ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 Pv(x) log − v 2 − 1 4 ( 1 y ) log v 2 + 1 4 ( 1 z ) √ x √ log ( 1 t ) log − 2x √ log ( 1 y )√ log( 1 z ) log( 1 t ) dxdydzdt = −i √ π 2 log(2) Proof. Use equation (10) and apply l’Hopital’s rule as k → −1 and simplify using equation (25.6.11) in [1]. R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 1113-1119 1118 Example 5. Apéry’s constant ζ(3), (12 ) ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 Pv(x) log − v 2 − 1 4 ( 1 y ) log v 2 + 1 4 ( 1 z ) √ x √ log ( 1 t ) log3 − 2x √ log ( 1 y )√ log( 1 z ) log( 1 t ) dxdydzdt = 3iζ(3) 16 √ 2π3/2 Proof. Use equation (10) and set k = −3 and simplify. Example 6. ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 Pv(x) log − v 2 − 1 2 ( 1 y ) log v 2 ( 1 z ) x log x √ log ( 1 y )√ log( 1 z ) log( 1 t )  ( xm log−m ( 1 t ) log m 2 ( 1 y ) log m 2 ( 1 z ) − xp log−p ( 1 t ) log p 2 ( 1 y ) log p 2 ( 1 z )) dxdydzdt = √ π ( 2−peiπpΦ ( e2ipπ, 1, π + i log(2) 2π ) − 2−meiπmΦ ( e2imπ, 1, π + i log(2) 2π )) (13) Proof. Use equation (7) and form a second equation by replacing m → p and taking their difference and setting k = −1, a = 1 and simplify. Example 7. ∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 Pv(x) log − v 2 − 3 8 ( 1 y ) log v 2 + 1 8 ( 1 z ) x3/4 √ log ( 1 t ) log x √ log ( 1 y )√ log( 1 z ) log( 1 t )  ( 4 √ x 8 √ log ( 1 y ) 8 √ log ( 1 z ) − 4 √ log ( 1 t )) dxdydzdt = √ π 2 ( 4 √ −2Φ ( i, 1, π + i log(2) 2π ) − iΦ ( −1, 1, π + i log(2) 2π )) (14) Proof. Use equation (13) and set p = 1/2,m = 1/4 and simplify. REFERENCES 1119 6. Discussion In this paper, we have presented a novel method for deriving a new integral involving the Legendre polynomial Pn(x) along with some interesting definite integrals using contour integration. The results presented were numerically verified for both real and imaginary and complex values of the parameters in the integrals using Mathematica by Wolfram. References [1] Nist digital library of mathematical functions. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. [2] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Else- vier/Academic Press, Amsterdam, seventh edition, 2007. [3] Iryna Fedotova Nina Virchenko. Generalized associated legendre functions and their applications. World Scientific, National Technical University of Ukraine, Singapore, 2001. [4] F. Oberhettinger. Tables of Mellin Transforms. Springer-Verlag Berlin Heidelberg, 1974. [5] Keith B. Oldham, Jan Myland, and Jerome Spanier. An Atlas of Functions: with Equator, the Atlas Function Calculator. Springer Science & Business Media, 07 2010. [6] Robert Reynolds and Allan Stauffer. A method for evaluating definite integrals in terms of special functions with examples. International Mathematical Forum, 15:235– 244, 2020.