2022/01/24 by Qianqian Cai, Cai, Qianqian, Su Hu +3
Mathematics · Physics and Astronomy · #11B68 #11M06 #11Y60 #26B20 #40G05 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2201.09674
openalex publication_date 2022/01/24 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
In this note, we extend Euler's transformation formula from the alternating series to more general series. Then we give new expressions for the Riemann zeta function ζ(s) by the generalized difference operator Δc, which provide analytic continuation of ζ(s) and new ways to evaluate the special values of ζ(-m) for m=0,1,2,…. Applying these results, we further extend Huylebrouck's generalization of Wallis' well-known formula for π in the half planes Re(s)>0 and Re(s)>-1, respectively. They imply several interesting special cases including \frac2π3(3)/(2)=\frac3(4)/(3)2(4)/(3) \frac2(1)/(3)⋅3(1)/(3)⋅3(1)/(3)⋅4(1)/(3)⋅6(2)/(3)⋅6(2)/(3)4(1)/(3)⋅4(1)/(3)⋅5(1)/(3)⋅5(1)/(3)⋅4(2)/(3)⋅5(2)/(3)⋯, 3γ-(log 3)/(2)=\frac3(1)/(3)⋅3(1)/(3)2(1)/(2)⋅4(1)/(4) \frac6(1)/(6)⋅6(1)/(6)5(1)/(5)⋅7(1)/(7)\frac9(1)/(9)⋅9(1)/(9)8(1)/(8)⋅10(1)/(10)⋯, and (3(\frac2πeγA12)2)(π2)/(18)=\frac3(1)/(32)⋅3(1)/(32)2(1)/(22)⋅4(1)/(42) \frac6(1)/(62)⋅6(1)/(62)5(1)/(52)⋅7(1)/(72)\frac9(1)/(92)⋅9(1)/(92)8(1)/(82)⋅10(1)/(102)⋯, where γ is the Euler-Mascheroni constant and A is the Glaisher-Kinkelin constant.