Sentence examples for functional equation for from inspiring English sources

We also give a functional equation for the generating function.

In 2006, Crabb [26] showed that Gessel's generalization of Miki's identity [27] is a direct consequence of a functional equation for the generating function.

In 1877, Dedekind gave, under the modular transformation, an elegant functional equation for the Dedekind eta function, which contains the Dedekind sums.

by following the lines of the first Riemann proof of the functional equation for the Riemann zeta function [15]: ζ ( 1 − s ) = 2 ( 2 π ) − s Γ ( s ) cos ( π s 2 ) ζ ( s ) (1.7).

A functional equation for the M-series is derived.

Now we introduce the following functional equation for fixed integers with : (1.6). in quasi-Banach spaces.

The multichannel scattering problem is formulated as a system of nonlinear functional equations for the wave function and reaction matrix.

In Table 1 we present a sequence of functional equations for the generating function (H x)=1+x), where OEIS means the On-Line Encyclopedia of Integer Sequences [5].

In the present paper we derive the functional equations for particular density functions and the moment equations for the system which are used in the investigation of solvability and mean square stability.

Now, we turn our attention to the following generation function defined in [1] since we need this generating function frequently to give some functional equations for a q-Eulerian number and polynomials attached to the character χ.

On the other hand, Berndt [1], Goldberg [2] and also Simsek [3] gave, under the modular transformation, other elegant functional equations for the theta functions, which contain six different arithmetic sums (Hardy-Berndt sums).