Normal solutions of difference equations, Euler?s functions and spirals

Abstract

We consider the linear first order difference equationfor the unknown function f : I ⊆ R → C. We present, under some conditions on the given function α, a product representation of a specific solution of the considered difference equation that can be understood as the normal or principal solution in the concept of N. E. Nörlund, F. John, W. Krull and others. This normal solution is characterized by its asymptotic behavior near infinity. In that way we get a characterization of e.g. the gamma function, the q-gamma function and of several classical spirals.

Difference equations, Euler’s summation formula and Hyers–Ulam stability

“…The proof is similar to that of Theorem 1 in Gronau [5], and is more or less standard. Let f be a solution of (15) which satisfies (20).…”

“…Herbert Kociemba [7] gave an approximation of (3) bỹ whereφ is an approximation of ϕ given by (5) ϕ(x) = −47/48 − 3 · π/8 + 1/(6 · √ x + 1) + 2 · √ x + 1.…”

“…c(x) = arctan √ x + 1, the function ϕ given by (5) is the normal solution of (4). From Theorem 2 we get the estimate |ϕ(x) − Φ(x)| = O(x −1/2−m ).…”

A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions

“…Considering higher values of p may provide inequalities that are tighter than (35). For instance, taking p = ¾, we obtain…”

“…Rohde [74] also generalized that result by modifying the convexity property. Gronau [35] proposed a variant of Krull's result and applied it to characterize the Euler beta and gamma functions and study certain spirals (see also Gronau [36]). Merkle and Ribeiro Merkle [60] proposed to combine Krull's result with differentiation techniques to characterize the Barnes G-function.…”