We give an elementary deduction of the Choi-Lee-Srivastava’s identities involving the Euler Mascheroni’s constant, thus from them is immediate the identity of Wilf

Aguilar, C. Hernández

Bonilla, Jose Luis Lopez

Vázquez, R. López

English

PURKH (E-Journals)

MathLAB Journal  Vol 1 No 3  (2018)                                                                              http://purkh.com/index.php/mathlab 
296 
Identities of Choi-Lee-Srivastava involving the Euler-Mascheroni’s constant 
C. Hernández-Aguilar, J. López-Bonilla, R. López-Vázquez, 
ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 5, 3er. Piso, Col. Lindavista CP 07738, CDMX, México;   
jlopezb@ipn.mx 
Abstract: 
 We give an elementary deduction of the Choi-Lee-Srivastava’s identities involving the Euler Mascheroni’s 
constant, thus from them is immediate the identity of Wilf.  
Subject Classification: 34A25, 34A26 
Keywords: Constant of Euler-Mascheroni, Gamma function, Newman-Weierstrass identity. 
1. Introduction 
   Wilf [1] proposed to prove the identity [2-4]: 
                                                 cosh (
𝜋
2
) =
𝜋
2
 𝑒𝛾  ∏  𝑒− 
1
𝑘∞𝑘=1  (1 +
1
𝑘
+
1
2 𝑘2
),                                                 (1) 
where 𝛾 =0.5772 1566 4901 5328 6060 … is the Euler-Mascheroni’s constant [4, 5]. 
In Sec. 2 we use properties of the gamma function [4-9] to give an elementary deduction of (1), and in Sec. 3 
this process allows generalize (1) to obtain the Choi-Lee-Srivastava’s expressions [2]: 
                            cosh(𝛼 𝜋) = 𝜋 (𝛼2 +
1
4
) 𝑒𝛾 ∏  𝑒
− 
1
𝑗∞
𝑗=1 (1 +
1
𝑗
+
𝛼2 + 
1
4
𝑗2
) ,            𝛼 ≠ ± 
𝑖
2
 ,                        (2) 
which implies (1) for 𝛼 =
1
2
 , and: 
                           sinh(𝛽 𝜋) = 𝛽 𝜋 (𝛽2 + 1) 𝑒2𝛾  ∏ 𝑒
− 
2
𝑗∞
𝑗=1  (1 +
2
𝑗
+
𝛽2 + 1
𝑗2
 ) ,       𝛽 ≠ ± 𝑖 .                        (3) 
2. Wilf’s formula 
   In [10] we find the following relation involving an infinite product and the gamma function: 
                                                    ∏
(𝑘 + 𝑧)2−𝑏
(𝑘 + 𝑧)2−𝑎
∞
𝑘=𝑚 =
Γ(𝑧 + 𝑚 − √𝑎) Γ(𝑧 + 𝑚 + √𝑎)
Γ(𝑧 + 𝑚 − √𝑏) Γ(𝑧 + 𝑚 + √𝑏)
 ,                                               (4) 
where we can employ 𝑎 = −𝑏 =
1
4
 , 𝑚 = 1  and  𝑧 =
1
2
  to obtain the expression: 
                            ∏
(2𝑘 + 1)2 + 1
(2𝑘 + 1)2 − 1
∞
𝑘=1 ≡ ∏  [1 +
1
2𝑘 (𝑘 + 1)
 ]∞𝑘=1 =
1
Γ(
3−𝑖
2
) Γ(
3+𝑖
2
)
=
2
Γ(
1−𝑖
2
) Γ(
1+𝑖
2
)
 .                            (5) 
   On the other hand, we know the property [7]: 
                                                         
𝜋
cosh (𝜋 𝑥)
= Γ (
1
2
− 𝑖 𝑥) Γ(
1
2
+ 𝑖 𝑥),                                                          (6) 
MathLAB Journal  Vol 1 No 3  (2018)                                                                              http://purkh.com/index.php/mathlab 
297 
that we can apply with  𝑥 =
1
2
  into (5) to deduce the identity: 
                                                        cosh (
𝜋
2
) =
𝜋
2
 ∏  [1 +
1
2𝑘 (𝑘 + 1)
 ]∞𝑘=1 .                                                       (7)  
Now we observe the relation: 
                                                              
1 +  
1
𝑘
  +  
1
2𝑘2
1 + 
1
 𝑘
= 1 +
1
2𝑘 (𝑘 + 1)
 ,                                                                (8) 
then (7) is equivalent to: 
                                                       cosh (
𝜋
2
) =
𝜋
2
  
∏ 𝑒
− 
1
𝑘∞
𝑘=1  (1 + 
1
𝑘
 + 
1
2𝑘2
)
∏  𝑒
− 
1
𝑟 ∞𝑟=1  (1 + 
1
𝑟
)
 ,                                                          (9) 
but the Newman (1848)-Weierstrass (1856) formula [7]: 
                                                         𝑧 𝑒𝛾𝑧  ∏  𝑒− 
𝑧
𝑟∞𝑟=1  (1 +
𝑧
𝑟
) =
1
Γ(𝑧)
 ,                                                          (10) 
with  𝑧 = 1 gives the expression: 
                                                             ∏ 𝑒− 
1
𝑟∞𝑟=1  (1 +
1
𝑟
) = 𝑒−𝛾 ,                                                                (11) 
whose application in (9) implies the Wilf’s identity (1) [1-4, 11], q.e.d. 
3. Choi-Lee-Srivasatava’s relations 
   The process indicated in Sec. 2 permits to prove (2) and (3), in fact, we use (4) with  𝑎 =
1
4
 , 𝑏 = −𝛼2 , 𝑚 = 1,
𝑧 =
1
2
 , and (6) for  𝑥 = 𝛼 ≠ ± 
𝑖
2
  : 
                                                           ∏
(2𝑘 + 1)2 + 4 𝛼2
(2𝑘 + 1)2 − 1
∞
𝑘=1 =
cosh (𝛼 𝜋)
𝜋 ( 𝛼2 +  
1
4
 )
 ,                                                       (12) 
however, we have the property: 
                                                             
(2𝑘 + 1)2 + 4 𝛼2
(2𝑘 + 1)2 − 1
=
1 +  
1
𝑘
  +  
𝛼2 +  
1
4
𝑘2
1 +  
1
𝑘
 ,                                                          (13) 
hence the application of (11) and (13) into (12) implies (2), q.e.d.  
   Similarly, from (4) for  𝑎 = 𝑚 = 𝑧 = 1, 𝑏 = −𝛽2, and the companion relation of (6) if  𝛽 ≠ ± 𝑖 ∶ 
                                                              
𝜋
sinh (𝜋 𝛽)
= 𝛽 Γ(𝑖𝛽) Γ(−𝑖𝛽),                                                            (14) 
we deduce the expression: 
                                                      ∏
(𝑘 + 1)2 + 𝛽2
(𝑘 + 1)2 − 1
∞
𝑘=1 =
2
𝛽 (𝛽2 + 1)
 
sinh (𝛽 𝜋)
𝜋
 ,                                                   (15) 
MathLAB Journal  Vol 1 No 3  (2018)                                                                              http://purkh.com/index.php/mathlab 
298 
but we have the decomposition: 
                                                                 
(𝑘 + 1)2 + 𝛽2
(𝑘 + 1)2 − 1
=
1 + 
2
𝑘
 + 
𝛽2 + 1
𝑘2
1 + 
2
𝑘
  ,                                                            (16) 
and from (10) with 𝑧 = 2: 
                                                               ∏ 𝑒− 
2
𝑟∞𝑟=1  (1 +
2
𝑟
) =
1
2
 𝑒−2 𝛾 ,                                                        (17) 
then the use of (16) and (17) into (15) gives (3), q.e.d. 
References 
1) H. S. Wilf, Problem 10588, Amer. Math. Monthly 104 (1997) 456. 
2)  J. Choi, J. Lee, H. M. Srivastava, A generalization of Wilf’s formula, Kodai Math. J. 26 (2003) 44-48. 
3)  Chao-Ping Chen, J. Choi, Two infinite product formulas with two parameters, Integral Transforms and  
4) Special Functions 24, No. 5 (2013) 357-363. 
5) H. M. Srivastava, J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier,  
6) London (2012). 
7) J. Havil, Gamma. Exploring Euler’s constant, Princeton University Press, New Jersey (2003). 
8) P. J. Davis, Leonhard Euler’s integral: A historical profile of the gamma function, Amer. Math.  
9) Monthly 66, No. 10 (1959) 849-869. 
10) E. Artin, The gamma function, Holt, Rinehart and Winston, New York (1964). 
11) G. Srinivasan, The gamma function: An eclectic tour, Amer. Math. Monthly 114, No. 4 (2007) 297-315. 
12) J. Bonnar, The gamma function, Treasure Trove of Mathematics (2017).  
13) http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html#InfinitexProducts 
14) J. Choi, T. Y. Seo, Evaluation of some infinite series, Indian J. Pure Appl. Math. 28 (1997) 791-796. 
 
 
 
 


Identities of Choi-Lee-Srivastava involving the Euler-Mascheroni’s constant

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Identities of Choi-Lee-Srivastava involving the Euler-Mascheroni’s constant

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