aa r X i v : . [ m a t h . G M ] A p r . ON A CERTAIN IDENTITY INVOLVING THE GAMMAFUNCTION
THEOPHILUS AGAMA
Abstract.
The goal of this paper is to prove the identity ⌊ s ⌋ X j =0 ( − j s j η s ( j ) + 1 e s − s s ⌊ s ⌋ X j =0 ( − j +1 α s ( j ) + (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19)(cid:18) ∞ X j = ⌊ s ⌋ +1 ( − j s j η s ( j ) + 1 e s − s s ∞ X j = ⌊ s ⌋ +1 ( − j +1 α s ( j ) (cid:19) = 1Γ( s + 1) , where η s ( j ) := (cid:18) e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (cid:19)(cid:18) s − js + ∞ X m =1 sm ( s + m ) − ∞ X m =1 s − jm ( s − j + m ) (cid:19) , and α s ( j ) := (cid:18) e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (cid:19)(cid:18) ∞ X m =1 sm ( s + m ) − ∞ X m =1 s − jm ( s − j + m ) (cid:19) , where Γ( s +1) is the Gamma function defined by Γ( s ) := ∞ R e − t t s − dt and γ =lim n −→∞ (cid:18) n P k =1 1 k − log n (cid:19) = 0 . · · · is the Euler-Mascheroni constant. INTRODUCTION
The Euler-Gamma function is defined by, Γ( s ) := ∞ R e − t t s − dt , valid in the entirecomplex plane, except at s = 0 , − , − , . . . where it has simple poles [1]. It can alsobe seen as a generalization of the factorial on the positive integers to the rationals.Indeed the Gamma function (See [2], [1]) satisfies the functional equation Γ(1) = 1 And here is the beginning of the second paragraph.
Date : August 28, 2018.2000
Mathematics Subject Classification.
Primary 54C40, 14E20; Secondary 46E25, 20C20.
Key words and phrases.
Gamma function, digamma function, poles. and Γ( s ) = Γ( s + n ) s ( s + 1)( s + 2) · · · ( s + n − s = 1 and n is a positive integer, then we have the expressionΓ( n + 1) = 1 · · · · n = n !. The Gamma function still remains valid for argumentsin the range − < s < s ) = Γ( s + 1) s . It also has the canonical product representation (See [3])Γ( s + 1) = e − γs ∞ Y m =1 (cid:18) mm + s (cid:19) e s/m , valid for s > −
1. The gamma function also has very key properties, most notablythe duplication and the complementary property (reflexive formula), which aregiven respectively as Γ( x )Γ(1 − x ) = π sin πx , and Γ( x )Γ( x + 1 /
2) = √ π x − Γ(2 x ) . For many more of these properties, the reader is encouraged to see [1]. The Gammafunction is also inextricably linked to some very interesting functions. Consider thedigamma function [1], the logarithmic derivative of the Gamma function defined byΨ( x ) := Γ ′ ( x )Γ( x ) = − γ + ∞ X m =1 ( x − m ( m + x − . The Gamma funtion has spawn a great deal of research and out of which has ledto the discovery of many beautiful identities and inequalities. More recently thegamma function has been studied by Alzer and many other authors. For moreresults on the gamma function, see [2], [3]. In this paper, however, we prove acertain identity related to the Gamma function.
N A CERTAIN IDENTITY INVOLVING THE GAMMA FUNCTION 3 MAIN THEOREM
Theorem 2.1.
For any s > , we have ⌊ s ⌋ X j =0 ( − j s j η s ( j ) + 1 e s − s s ⌊ s ⌋ X j =0 ( − j +1 α s ( j ) + (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19)(cid:18) ∞ X j = ⌊ s ⌋ +1 ( − j s j η s ( j ) + 1 e s − s s ∞ X j = ⌊ s ⌋ +1 ( − j +1 α s ( j ) (cid:19) = 1Γ( s + 1) , where η s ( j ) := (cid:18) e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (cid:19)(cid:18) s − js + ∞ X m =1 sm ( s + m ) − ∞ X m =1 s − jm ( s − j + m ) (cid:19) , and α s ( j ) := (cid:18) e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (cid:19)(cid:18) ∞ X m =1 sm ( s + m ) − ∞ X m =1 s − jm ( s − j + m ) (cid:19) , where Γ( s + 1) is the Gamma function defined by Γ( s ) := ∞ R e − t t s − dt and γ =lim n −→∞ (cid:18) n P k =1 1 k − log n (cid:19) = 0 . · · · is the Euler-Mascheroni constant.Proof. Let f ( t ) be a real-valued function, contineously differentiable on the interval[0 , ∞ ) and f ( t ) ≥ t ∈ [0 , ∞ ). Then we set F ( s ) := s Z f ( t ) (cid:18) log f ( t ) (cid:19) s dt for s >
1. In the simplest case, we choose f ( t ) = e t , since it satisfies the hypothesis.Thus F ( s ) = s R e t t s dt . By application of integration by parts, we find that F ( s ) := s R e t t s = e s s s − se s s s − + s ( s − e s s s − − s ( s − s − e s s s − + s ( s − s − s − e s s s − + I ( s ) + β ( s ), where I ( s ) and β ( s ) are convergent. More precisely, we can THEOPHILUS AGAMA write F ( s ) in a closed form as F ( s ) = ⌊ s ⌋ X j =0 ( − j e s s s − j Γ( s + 1)Γ( s + 1 − j ) + e ⌊ s ⌋ X j =0 ( − j +1 Γ( s + 1)Γ( s + 1 − j )+ (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19)(cid:18) ∞ X j = ⌊ s ⌋ +1 ( − j e s s s − j Γ( s + 1)Γ( s + 1 − j )+ e ∞ X j = ⌊ s ⌋ +1 ( − j +1 Γ( s + 1)Γ( s + 1 − j ) (cid:19) . Now, since Γ( s ) is analytic in the half plane Re( s ) ≥
1, it follows by the convergenceof F ( s ) that F ′ ( s ) = e s s s Γ( s + 1) ⌊ s ⌋ X j =0 ( − j s j Γ( s + 1 − j ) + e s s s (log s + 1)Γ( s + 1) ⌊ s ⌋ X j =0 ( − j s j Γ( s + 1 − j ) + e s s s Γ ′ ( s + 1) ⌊ s ⌋ X j =0 ( − j s j Γ( s + 1 − j ) + e s s s Γ( s + 1) ⌊ s ⌋ X j =0 ( − j +1 js j − Γ( s + 1 − j ) + s j Γ ′ ( s + 1 − j ) s j Γ ( s + 1 − j )+ (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19) e s s s Γ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j s j Γ( s + 1 − j ) + (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19) e s s s (log s + 1)Γ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j s j Γ( s + 1 − j )+ (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19) e s s s Γ ′ ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j s j Γ( s + 1 − j )+ (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19) e s s s Γ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j +1 js j − Γ( s + 1 − j ) + s j Γ ′ ( s + 1 − j ) s j Γ ( s + 1 − j )+Γ ′ ( s + 1) ⌊ s ⌋ X j =0 ( − j +1 s + 1 − j ) + Γ( s + 1) ⌊ s ⌋ X j =0 ( − j +2 Γ ′ ( s + 1 − j )Γ ( s + 1 − j )+ (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19)(cid:18) Γ ′ ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j +1 s + 1 − j )+Γ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j +2 Γ ′ ( s + 1 − j )Γ ( s + 1 − j ) (cid:19) . N A CERTAIN IDENTITY INVOLVING THE GAMMA FUNCTION 5
On the other hand F ′ ( s ) = e s s s . Arranging terms and comparing both results wefind that 1 = ⌊ s ⌋ X j =0 ( − j (cid:18) Γ( s + 1) s j Γ( s + 1 − j ) + (log s + 1)Γ( s + 1) s j Γ( s + 1 − j ) + Γ ′ ( s + 1) s j Γ( s + 1 − j ) − j Γ( s + 1) s j +1 Γ( s + 1 − j ) − Γ( s + 1)Γ ′ ( s + 1 − j ) s j Γ ( s + 1 − j ) (cid:19) + (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19) ∞ X j = ⌊ s ⌋ +1 ( − j (cid:18) Γ( s + 1) s j Γ( s + 1 − j )+ (log s + 1)Γ( s + 1) s j Γ( s + 1 − j ) + Γ ′ ( s + 1) s j Γ( s + 1 − j ) − j Γ( s + 1) s j +1 Γ( s + 1 − j ) − Γ( s + 1)Γ ′ ( s + 1 − j ) s j Γ ( s + 1 − j ) (cid:19) + 1 e s − s s Γ ′ ( s + 1) ⌊ s ⌋ X j =0 ( − j +1 s + 1 − j ) + 1 e s − s s Γ( s + 1) ⌊ s ⌋ X j =0 ( − j +2 Γ ′ ( s + 1 − j )Γ ( s + 1 − j )+ (cid:18) − (( − s −⌊ s ⌋ +2 ) / ( s −⌊ s ⌋ +2) (cid:19)(cid:18) e s − s s Γ ′ ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j +1 s + 1 − j )+ 1 e s − s s Γ( s + 1) ∞ X j = ⌊ s ⌋ +1 ( − j +2 Γ ′ ( s + 1 − j )Γ ( s + 1 − j ) (cid:19) (2.1)Using the following identities involving the Gamma function [1]1Γ( s + 1 − j ) := e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (2.2) Γ ′ ( s + 1)Γ( s + 1) := − γ + ∞ X m =1 sm ( s + m ) , (2.3)the remaining task is to arrange the terms and apply these identities and identify thefunction η s ( j ) and α s ( j ). We leave the remaining task to the reader to verify. (cid:3) Remark . Now we examine some immediate conequences of the above result, inthe following sequel.
Corollary 1.
The identity ∞ X j =0 ( − j (cid:18) (cid:19) j η / ( j ) + 1(3 / / √ e ∞ X j =0 ( − j +1 α / ( j ) = 43 √ π , THEOPHILUS AGAMA where η / ( j ) = (cid:18) e γ (3 / − j ) ∞ Y m =1 (cid:18) / − jm (cid:19) e − (3 / − j ) /m (cid:19)(cid:18) (cid:18) / (cid:19) − j
3+ 32 ∞ X m =1 m ( m + 3 / − ∞ X m =1 32 − jm ( m + 3 / − j ) (cid:19) , and α / ( j ) := (cid:18) e γ (3 / − j ) ∞ Y m =1 (cid:18) / − jm (cid:19) e − / − jm (cid:19)(cid:18) ∞ X m =1 / m (3 / m ) − ∞ X m =1 / − jm (3 / − j + m ) (cid:19) , remains valid. Proof.
Let us set s = in Theorem 2.1. Then it follows that ∞ X j =0 ( − j (cid:18) (cid:19) j η / ( j ) + 1(3 / / √ e ∞ X j =0 ( − j +1 α / ( j ) = 1Γ(5 /
2) = 43Γ(1 /
2) = 43 √ π , where we have used the relation Γ( ) = √ π [1]. The proof is completed by com-puting η / ( j ) and α / ( j ) given in Theorem 2.1. (cid:3) Corollary 2.
The identity ∞ X j =0 ( − j (cid:18) (cid:19) j η / ( j ) + 1(5 / / √ e ∞ X j =0 ( − j +1 α / ( j ) = 910Γ(2 / , is valid, where η / ( j ) = (cid:18) e γ (5 / − j ) ∞ Y m =1 (cid:18) / − jm (cid:19) e − (5 / − j ) (cid:19)(cid:18) (cid:18) / (cid:19) − j ∞ X m =1 / m (5 / m ) − ∞ X m =1 / − jm (5 / − j + m ) (cid:19) and α / ( j ) := (cid:18) e γ (5 / − j ) ∞ Y m =1 (cid:18) / − jm (cid:19) e − / − jm (cid:19)(cid:18) ∞ X m =1 / m (5 / m ) − ∞ X m =1 / − jm (5 / − j + m ) (cid:19) , Proof.
The result follows by setting s = in Theorem 2.1, and computing η / ( j )and α / ( j ). (cid:3) N A CERTAIN IDENTITY INVOLVING THE GAMMA FUNCTION 7
Corollary 3.
For any integer s ≥
2, the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X j =0 ( − j η s ( j ) s j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e s − s s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X j =0 ( − j α s ( j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s + 1) < s X j =0 | η s ( j ) | + | α s ( j ) | s j , where η s ( j ) := (cid:18) e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (cid:19)(cid:18) s − js + ∞ X m =1 sm ( s + m ) − ∞ X m =1 s − jm ( s − j + m ) (cid:19) and α s ( j ) := (cid:18) e γ ( s − j ) ∞ Y m =1 (cid:18) s − jm (cid:19) e − ( s − j ) /m (cid:19)(cid:18) ∞ X m =1 sm ( s + m ) − ∞ X m =1 s − jm ( s − j + m ) (cid:19) , is valid. Proof. If s ≥ s X j =0 ( − j s j η s ( j ) − e s − s s s X j =0 ( − j α s ( j ) = 1Γ( s + 1) , and the result follows immediately by applying the triangle inequality. (cid:3) References
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Introduction to the gamma function , American Journalof Scientific Research, 2002.2. Batir, Necdet,
Bounds for the gamma function , arXiv preprint arXiv:1705.06167, 2015.3. Nantomah, Kwara and Prempeh, Edward and Twum, S. Boakye,
Some inequalities for the q -Extension of the Gamma Function , arXiv preprint arXiv:1510.03459, 2015. Department of Mathematics, African Institute for Mathematical science, Ghana
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