A generalization of Krull-Webster's theory to higher order convex functions: multiple Γ-type functions
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We provide uniqueness and existence results for the eventually p-convex and eventually p-concave solutions to the difference equation Δ f=g on the open half-line (0,∞), where p is a given nonnegative integer and g is a given function satisfying the asymptotic property that the sequence n↦Δ^p g(n) converges to zero. These solutions, that we call logΓ_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 logΓ_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.
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