Reference : A generalization of Krull-Webster's theory to higher order convex functions: multiple...
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A generalization of Krull-Webster's theory to higher order convex functions: multiple gamma-type functions
Marichal, Jean-Luc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Zénaïdi, Naïm [University of Liège, Department of Mathematics, Liège, Belgium]
[en] Difference equation ; Higher order convexity ; Bohr-Mollerup-Artin's theorem ; Krull-Webster's theory ; Generalized Stirling's formula ; Generalized Stirling's constant ; Generalized Euler's constant ; Euler's reflection formula ; Euler's infinite product ; Weierstrass' infinite product ; Gauss' multiplication theorem ; Gauss' digamma theorem ; Raabe's formula ; Wallis's product formula ; Fontana-Mascheroni's series ; Barnes G-function ; Hurwitz zeta function ; Gamma-related function ; Multiple gamma-type function ; Generalized Stieltjes constant
[en] We provide uniqueness and existence results for the eventually $p$-convex and eventually $p$-concave solutions to the difference equation $\Delta f=g$ on the open half-line $(0,\infty)$, where $p$ is a given nonnegative integer and $g$ is a given function satisfying the asymptotic property that the sequence $n\mapsto\Delta^p g(n)$ converges to zero. These solutions, that we call $\log\Gamma_p$-type functions, include various special functions such as the polygamma functions, the logarithm of the Barnes $G$-function, and the Hurwitz zeta function. Our results generalize to any nonnegative integer $p$ the special case when $p=1$ obtained by Krull and Webster, who both generalized Bohr-Mollerup-Artin's characterization of the gamma function.

We also follow and generalize Webster's approach and provide for $\log\Gamma_p$-type functions analogues of Euler's infinite product, Weierstrass' infinite product, Gauss' limit, Gauss' multiplication formula, Legendre's duplication formula, Euler's constant, Stirling's constant, Stirling's formula, Wallis's product formula, and Raabe's formula for the gamma function. We also introduce and discuss analogues of Binet's function, Burnside's formula, Fontana-Mascheroni's series, Euler's reflection formula, and Gauss' digamma theorem.

Lastly, we apply our results to several special functions, including the Hurwitz zeta function and the generalized Stieltjes constants, and show through these examples how powerful is our theory to produce formulas and identities almost systematically.
University of Luxembourg - UL
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