Limiting Values and Functional and Difference Equations

Abstract

Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients.

Arithmetical Fourier transforms and Hilbert space: Restoration of the lost legacy

“…One of the most prominent cases of applications of DFT to number theory is [Leh75] and subsequent [Fu90], both of which use (3.51) and have been incorporated as special cases of a more general theorem in [WAK20].…”

“…As has been wellestablished cf. e. g. [WAK20], the Lerch zeta-function being a Fourier series as well as a zeta-function that satisfies the functional equation, turns out to be quite useful.…”

On some Lambert-like series

“…Concluding remarks: Lehmer [Leh75] made use of both real and imaginary parts of the monologuefunction (1.3) in connection with his study on generalized Euler constants, cf. [WAK20]. There we state a general principle to the effect that Laurent coefficients is that which governs the situation including Kronecker limit type formulas.…”