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Gamma , Psi , Bernoulli Functions Via Hurwitz Zeta Function

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Gamma , Psi , Bernoulli Functions via Hurwitz Zeta Function . VIVEK V . RANE A-3/203 , Anand Nagar , Dahisar (East) Mumbai – 400 068 INDIA e-mail address : v_v_rane@ yahoo.co.in Abstract : Using three basic facts concerning Hurwitz zeta function , we give new natural proofs of the known results on Bernoulli polynomials , gamma function and also obtain Gauss’ expression for psi function at a rational point all in a unified fashion . We also give a new proof of the relation between log gamma and the derivative of the Hurwitz zeta function including that of Stirling’s expression for log gamma . Keywords : Hurwitz zeta function , Gamma function , Dirichlet character , Dirichlet L-series , Bernoulli polynomials/Numbers , Gauss’ psi function . Gamma , Psi , Bernoulli Functions via Hurwitz Zeta Function VIVEK V . RANE A-3/203 , Anand Nagar , Dahisar (East) Mumbai – 400 068 INDIA e-mail address : v_v_rane@ yahoo.co.in The object of this paper is to indicate that the theory of Euler’s gamma function , that of Gauss ‘ psi function and that of Bernoulli functions can be developed naturally , with ease from the theory of Hurwitz’s zeta function and its derivatives . For a complex variable s and for a complex number α≠0 , -1, -2 , ………. let ζ(s,α ) be the ∞ −푠 Hurwitz’s zeta function defined by ζ(s,α)= 푛=0(푛+∝) for Re s>1 ; and its analytic continuation . Let ζ(s,1)=ζ(s) be the Riemann zeta function . If χ(mod q) is a Dirichlet character modulo an integer q≥1 and 푎 if L(s,χ) is the corresponding Dirichlet L-series , then in view of L(s,χ)=q-s 푞 휒 푎 휁(푠, ) , it is already 푎=1 푞 clear from author [2] and many other works of the author that the theory of Dirichlet L-series can be developed naturally from the theory of ζ(s,α) . For integers n≥0 , let Bn(α) be the Bernoulli polynomials in α defined by ∝푧 푛 푧푒 ∞ 푧 = 퐵 (훼) for |z|<2π ; and let B =B (0) be Bernoulli numbers . Note that B (α)=1 . Then it is 푒 푧−1 푛=0 푛 푛! n n 0 known that if n≥0 is an integer , Bn+1(α)=(-n-1)ζ(-n,α) and this fact has been proved in author [4] by comparing the Fourier series expressions of ζ(-n,α) and Bn+1(α) . As a function of s , ζ(s,α) is a meromorphic function with a simple pole at s=1 with residue 1 . As a function of α , ζ(s,α) is an analytic function of α except possibly for α=0,-1,-2 , ………. However , if m≥0 is an integer , then ζ(-m,α) is a polynomial in α and thus is an entire function of α (See author [2]) . For any integer r≥0 , we write : 2 : 휕푟 휕 ζ(r)(s,α)= 휁(푠, 훼) . In particular , ζ’(s,α)= ζ(s,α) . As a function of α , 휁(푟)(s,α) is an analytic function 휕푠푟 휕푠 of α except possibly for α=0 , -1 , -2 , ………. (See author [2] , author [3] .) If s≠1 and α≠0 , -1 , -2, ………., 휕푟1 휕푟2 휕푟2 휕 푟1 then 휁 푠, 훼 = 휁 푠, 훼 , where 푟 푟 ≥ 0 are integers . See for example author [3] . 휕∝푟1 휕푠푟2 휕푠푟2 휕∝푟1 1 , 2 휕 휕 휕 휕 In particular , if s≠1 and if α≠0 ,-1 , -2 , ………. , then 휁 푠, 훼 = 휁 푠, 훼 . 휕훼 휕푠 휕푠 휕훼 Next , let Г(α) be the Euler’s gamma function defined by Weierstrass ‘ product namely 훼 1 훾훼 훼 − Г(훼) = 훼푒 Π (1+ )푒 푛 , where γ is Euler’s constant . Then it is known that ζ’(0,α)=log . In Г(훼) n≥`1 푛 2휋 what follows , this fact will be proved with ease afresh as Proposition 5 , wherein we shall also obtain 푑 푑 Г(훼) 휕 Stirling’s expression for log Г(α) . Next , let ψ(α)= log Г(α)= log = ζ’(0,α) . In Proposition 4 푑훼 푑훼 2휋 휕훼 푎 below , we shall obtain Gauss’ expression for ψ , where 1≤a<q are integers . 푞 We shall show that by defining 1) 퐵푛+1 훼 = −푛 − 1 휁(−푛, 훼) for n≥ -1 휕 2) log Г(α)= ζ(s,α) at s=0 휕푠 휕 휕 3) ψ(α)= 휁 푠, 훼 at s=0 휕훼 휕푠 we can develop naturally the theory of Bernoulli polynomials , gamma function and Gauss’ ψ function in terms of ζ(s,α) and its derivatives as in the case of Dirichlet L-series . Basically , there are three facts concerning ζ(s,α) . 휕 1) 휁 푠, 훼 = −푠휁(푠 + 1 , 훼) 휕훼 : 3 ∶ 2) 휁 푠, 훼 − 휁 푠, 훼 + 1 = 훼−푠 휋푠 3) 퐹표푟 푅푒 s<1 and for 0<α<1 , ζ(s,α)=2푠휋푠−1Г(1-s) sin + 2휋푛훼 푛푠−1 푛≥1 2 2휋푛훼 s-1 =Г(1-s) |푛|≥1 푒 (2πin) Fact I) resulted in the Taylor series expansion of ζ(s,α+1) in author *1+ in the form (−훼)푛 휁 푠, 훼 + 1 = 휁 푠 + s(s+1) ………. (s+n-1)ζ(s+n) and consequently in author [2] , we find 푛≥1 푛! 훼푚 +1 that for integral m≥0 , ζ(-m,α)= 푚 푚 ζ(-k)훼푚−푘 + 훼푚 − . In author [4] , it has been shown 푘=0 푘 푚 +1 푚′ 푚′ 푚 ′ −푘 that the above expression for ζ(-m,α) with m≥0 , is equivalent to 퐵푚′ (α)= 푘=0 푘 퐵푘 훼 푤푡ℎ 푚′ = 푚 + 1 . Thus fact II) and fact III) remain to be exploited . Using these facts , we prove with ease the following four propositions . Proposition 1 We have 푚−1 1) 퐵푚 (α+1) - 퐵푚 훼 = 푚훼 for m≥0 . 2) Г 훼 + 1 = 훼Г 훼 1 Corollary : ψ(α+1)=ψ (α) + 훼 Proposition 2 : We have 푚 1) 퐵푚 1 − 훼 = −1 퐵푚 (훼) 휋 2) Γ 훼 Γ 1 − 훼 = 푠푛휋훼 Corollary : ψ(1-α)=ψ (α) +π cot πα Proposition 3 : We have for any integer m≥2 and for any integer n≥1 , : 4 : 1 2 푚 −1 1) 퐵 훼 + 퐵 훼 + + 퐵 (훼 + ) + ⋯ … … . +퐵 훼 + = 푚1−푛 퐵 푚훼 푛 푛 푚 푛 푚 푛 푚 푛 푛−1 1 1 푛−1 −푛훼 2) Γ 훼 Γ 훼 + … … … . Γ 훼 + = 2휋 2 푛2 Γ(푛훼) 푛 푛 푘 Corollary : We have 푚 −1 sinπ⁡(훼 + ) = 21−푚 sin 휋푚훼 . 푘=0 푚 Note that in the above three propositions , the results are in pairs . The results pertaining to Bernoulli polynomials are about sums and the results pertaining to gamma function are about products . Next , we state remaining Propositions . Proposition 4 : We have If 1≤a<q are integers , 푎 2휋푟푎 휋푟 휋 2휋푟푎 then 휓 = −(훾 + log 푞 ) + 푞−1 cos log 2 sin + 푞−1 푟 sin 푞 푟=1 푞 푞 푞 푟=1 푞 −1 Next with Weierstrass product definition of Г(α) and using Euler’s summation formula , we 1 state and prove Proposition 5 . In what follows , we shall write 휙 푢 = 푢 − 푢 − , where [u] denotes 2 integral part of the real variable u . 1 Proposition 5 : Let 휙 푢 = 푢 − 푢 − . 2 ∞ 휙 (푢) ∞ 휙(푢) ′ Γ(훼) Then log Г(α)=(α-1/2) log α – α +1 + 푑푢 − du and 휁 0, 훼 = log . 1 푢 0 푢+훼 2휋 Next , we prove our propositions . We shall prove Proposition 5 first . 1 Proof of Proposition 5 : From Weierstrass product expression for on taking logarithm , Γ(훼) 훼 훼 we have log Г 훼 = ( − log⁡(1 + )) − 훾훼 − log⁡ 훼 . 푛≥1 푛 푛 : 5 : On using Euler’s summation formula 1 ∞ 훼 훼 log Γ( 훼) = 훼 − log 1 + 훼 − 훾훼 − log 훼 + ( − log⁡(1 + ))du + 2 1 푢 푢 ∞ 푑 훼 훼 + ∅(푢) ( - log(1+ ))푑푢 , where 휙 푢 = 푢 − 푢 − 1/2 . 1 푑푢 푢 푢 ∞ 훼 훼 푢 ∞ 푢 1 Next ( − log⁡(1 + ))du = 푢 + 훼 푙표 = lim푢 ∞ 푢 + 훼 푙표 − 1 + 훼 푙표 1 푢 푢 푢+훼 푢=1 푢+훼 1+훼 = -훼 + 1 + 훼 log 1 + 훼 . 1 1 ∞ 푑 훼 훼 Thus log Γ 훼 = (훼 + ) log (1+α) – (γ + ) α – log α + 휙 푢 − log 1 + 푑푢 . 2 2 1 푑푢 푢 푢 ∞ 푑 훼 훼 ∞ ∅(푢) 1 1 푁푒푥푡 , 휙(푢) − log⁡(1 + ) du = 훼 − du 1 푑푢 푢 푢 1 푢 푢+훼 푢 ∞ 1 1 ∞ ∅(푢) ∞ ∅(푢) ∞ ∅(푢) 1 = 휙(푢) − 푑푢 − 훼 푑푢 = du - du +α 훾 − 1 푢 푢+훼 1 푢2 1 푢 1 푢+훼 2 ∞ ∅(푢) ∞ ∅(푢) 1 1 1 = du - du +α 훾 − + 1 +(훼 + ) log α – (α+ )log(1+α) . 1 푢 0 푢+훼 2 2 2 1 ∞ ∅(푢) ∞ ∅(푢) Thus log Γ 훼 = (훼 − ) log α – α+1 + du - du . 2 1 푢 0 푢+훼 ∞ ∅(푢) This gives an expression for log Γ(훼) . Note that integrating by parts repeatedly , can be 0 푢+훼 developed into an asymptotic series in α . ′ Γ(훼) Next , we shall prove 휁 0, 훼 = log . For 0<α≤1 , Re s>1 and for arbitrary x>0 , 2휋 −푠 −푠 −푠 we have 휁 푠, 훼 = 푛≥0(푛 + 훼) = 0≤푛≤푥−훼(푛 + 훼) + 푛>푥−훼(푛 + 훼) . −푠 We use Euler’s summation for 푛>푥−훼(푛 + 훼) . : 6 : This gives for x>0 , 0<α≤1 and for Re s>0 , 1−푠 −푠 푥 1 −푠 ∞ 휙 푢−훼 휁 푠, 훼 = (푛 + 훼) + + (x-α-[x-α+- )푥 -s du .
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    ALPHA PSI OMEGA THE NATIONAL THEATRE HONOR SOCIETY Its Aims and Purpose ALPHA PSI OMEGA was organized as a theatre honor society for the purpose of providing acknowledgement to those demonstrating a high standard of accomplishment in theatre and, through the expansion of ALPHA PSI OMEGA among colleges and universities, providing a wider fellowship for those interested in theatre. The society is not intended to take the place of any regular theatre clubs or producing groups, but as students qualify they may be rewarded by election to membership in this society. Revised August 24, 2016 * * * * * * * * * * CONSTITUTION AND BY-LAWS OF THE APLHA GAMMA ETA CHAPTER of THE ALPHA PSI OMEGA NATIONAL THEATRE HONOR SOCIETY Preamble We, the members of ALPHA PSI OMEGA, in order to develop talents in all aspects of theatre, to foster the cultural values we believe theatre develops, and to encourage cooperation and collaboration among member chapters, do hereby form and establish this constitution of THE ALPHA GAMMA ETA CHAPTER of THE ALPHA PSI OMEGA NATIONAL THEATRE HONOR SOCIETY. Article 1 Purpose and Jurisdiction The purpose of this, the Alpha Gamma Eta Chapter of ALPHA PSI OMEGA, is to stimulate interest in theatre activities at Saginaw Valley State University and to secure for the university all the advantages and mutual helpfulness provided by a large national honor society. By electing students to membership, the society acts as a reward for their participation in theatre activities of the university. This chapter is not intended to take the place of any existing theatre organization at the university. This chapter agrees to follow all constitutional laws and by-laws listed herein.
  • On Some Series Representations of the Hurwitz Zeta Function Mark W

    On Some Series Representations of the Hurwitz Zeta Function Mark W

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 216 (2008) 297–305 www.elsevier.com/locate/cam On some series representations of the Hurwitz zeta function Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, USA Received 21 November 2006; received in revised form 3 May 2007 Abstract A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants. © 2007 Elsevier B.V. All rights reserved. MSC: 11M06; 11M35; 33B15 Keywords: Hurwitz zeta function; Riemann zeta function; Polygamma function; Lerch zeta function; Series representation; Integral representation; Generalized harmonic numbers 1. Introduction (s, a)= ∞ (n+a)−s s> a> The Hurwitz zeta function, defined by n=0 for Re 1 and Re 0, extends to a meromorphic function in the entire complex s-plane. This analytic continuation to C has a simple pole of residue one. This is reflected in the Laurent expansion ∞ n 1 (−1) n (s, a) = + n(a)(s − 1) , (1) s − 1 n! n=0 (a) (a)=−(a) =/ wherein k are designated the Stieltjes constants [3,4,9,13,18,20] and 0 , where is the digamma a a= 1 function.
  • L-Series and Hurwitz Zeta Functions Associated with the Universal Formal Group

    L-Series and Hurwitz Zeta Functions Associated with the Universal Formal Group

    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 133-144 L-series and Hurwitz zeta functions associated with the universal formal group PIERGIULIO TEMPESTA Abstract. The properties of the universal Bernoulli polynomials are illustrated and a new class of related L-functions is constructed. A generalization of the Riemann-Hurwitz zeta function is also proposed. Mathematics Subject Classification (2010): 11M41 (primary); 55N22 (sec- ondary). 1. Introduction The aim of this article is to establish a connection between the theory of formal groups on one side and a class of generalized Bernoulli polynomials and Dirichlet series on the other side. Some of the results of this paper were announced in the communication [26]. We will prove that the correspondence between the Bernoulli polynomials and the Riemann zeta function can be extended to a larger class of polynomials, by introducing the universal Bernoulli polynomials and the associated Dirichlet series. Also, in the same spirit, generalized Hurwitz zeta functions are defined. Let R be a commutative ring with identity, and R {x1, x2,...} be the ring of formal power series in x1, x2,...with coefficients in R.Werecall that a commuta- tive one-dimensional formal group law over R is a formal power series (x, y) ∈ R {x, y} such that 1) (x, 0) = (0, x) = x 2) ( (x, y) , z) = (x,(y, z)) . When (x, y) = (y, x), the formal group law is said to be commutative. The existence of an inverse formal series ϕ (x) ∈ R {x} such that (x,ϕ(x)) = 0 follows from the previous definition.