2021/09/03 by Su Hu, Min-Soo Kim, Hu, Su +1
Mathematics · #11M35 #33B15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2109.01477
openalex publication_date 2021/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ(z) be the modified gamma function introduced by the authors in a recent preprint "arXiv2106.14674". In this note, we obtain the following Mizuno-type result: ∏m=0∞\∏j=1n(m+zj)\^(-1)m=\frac(√\fracπ2)n∏j=1nΓ(zj), which imply a Kurokawa--Wakayama type formula ∏m=0^∞((m+x)n-yn)^(-1)m =\frac(√\fracπ2)n∏ζn=1Γ(x-ζy) and a Lerch-type formula ∏m=0^∞(m+x)^(-1)m=\frac√\fracπ2Γ(x). By setting x=1 in the above result, we recover Wallis' 1656 fomula (2⋅2)/(1⋅ 3)(4⋅4)/(3⋅ 5)(6⋅6)/(5⋅ 7)⋯=\fracπ2.