Abstract. This paper presents a new method to solve functional equations of multivariate generating functions, such as F (r, s) = e(r, s)+xf(r, s)F (1, 1)+xg(r, s)F (qr, 1)+ xh(r, s)F (qr, qs), giving a formula for F (r, s) in terms of a sum over finite sequences. We use this method to show how one would calculate the coefficients of the generating function for parallelogram polyominoes, which ...

The first part of this writeup gives Riemann’s argument that the completion of zeta, Z(s) = π−s/2Γ(s/2)ζ(s), Re(s) > 1 has a meromorphic continuation to the full s-plane, analytic except for simple poles at s = 0 and s = 1, and the continuation satisfies the functional equation Z(s) = Z(1− s), s ∈ C. The continuation is no longer defined by the sum. Instead, it is defined by a wellbehaved integ...

The first part of this writeup gives Riemann’s argument that the closely related function Z(s) = π−s/2Γ(s/2)ζ(s), Re(s) > 1 has a meromorphic continuation to the full s-plane, analytic except for simple poles at s = 0 and s = 1, and the continuation satisfies the functional equation Z(s) = Z(1− s), s ∈ C. The continuation is no longer defined by the sum. Essentially the same ideas apply to Diri...

The first part of this writeup gives Riemann’s argument that the completion of zeta, Z(s) = π−s/2Γ(s/2)ζ(s), Re(s) > 1 has a meromorphic continuation to the full s-plane, analytic except for simple poles at s = 0 and s = 1, and the continuation satisfies the functional equation Z(s) = Z(1− s), s ∈ C. The continuation is no longer defined by the sum. Instead, it is defined by a wellbehaved integ...

The aim of this paper is to discuss the asymptotic properties of the coefficients of generating functions which satisfy a system of functional equations. It turns out that under certain general conditions these coefficients are related to the distribution of a multivariate random variable that is asymptotically normal. As an application it turns out that the distribution of the terminal symbols...

in this paper, we consider orthogonal stability of mixed type additive and cubic functional equation of the form $$f(2x+y)+f(2x-y)-f(4x)=2f (x+y)+2f(x-y)-8f(2x) +10f(x)-2f(-x),$$ with $xbot y$, where $bot$  is orthogonality in the sense of ratz.

Two functional equations are considered that are motivated by three considerations: work in utility theory and psychophysics, questions concerning when pairs of degree 1 homogeneous functions can be homomorphic and calculating their homomorphisms, and the link of the latter questions to quasilinear mean values. The first equation is h(σ(y)x + [1 - σ(y)]y) = τ(y)h(x) + [1 - τ(y)]h(y) (x ≥ y ≥ 0)...

The convergence of waveform relaxation techniques for solving functional-diierential equations is studied. New error estimates are derived that hold under linear and nonlinear conditions for the right-hand side of the equation. Sharp error bounds are obtained under generalized time-dependent Lipschitz conditions. The convergence of the waveform method and the quality of the a priori error bound...

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As id="M...