Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

Summary The multidimensional (partial) difference equation with periodical coefficients is transformed into an equation for a vector sequence. Integral formulae for the vector fundamental solution are developed and some results about its asymptotic properties are explained. As an example, the results are used for a simple difference equation on a hexagonal grid.     

On the peripheral spectrum of monic operator polynomials with positive coefficients

I wish to thank R. Nagel for his guidance and suggestions in the preparation of this paper. Also, I would like to thank G. Greiner and F. Räbiger for many interesting and helpful discussions.     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

On solving certain differential equations with variable coefficients

We show how to solve certain types of linear ordinary differential equations with variable coefficients by using Appell polynomials.     

On congruences related to central binomial coefficients

It is known that

     

On approximate solutions of the Pexider equation

Summary LetX be an abelian (topological) group andY a normed space. In this paper the following functional inequality is considered: {ie143-1} This inequality is a similar generalization of the Pexider equation as J. Tabor's generalization of the Cauchy equation (cf. [3], [4]). The solutions of our inequality have similar properties as the solutions of the Pexider equation. Continuity and related properties of the solutions are investigated as well.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we have

     

Linear recurrence relations for binomial coefficients modulo a prime

We investigate when the sequence of binomial coefficients modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0?i?k. In particular, we prove that this cannot occur if 2h?k<p−h. This hypothesis can be weakened to 2h?k<p if we assume, in addition, that the characteristic polynomial of the relation does not have −1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.     

Extrapolation with spline-collocation methods for two-point boundary-value problems I: Proposals and justifications

Asymptotic expansions for the error in some spline interpolation schemes are used to derive asymptotic expansions for the truncation errors in some spline-collocation methods for two-point boundary-value problems. This raises the possibility of using Richardson extrapolation or iterated deferred corrections to develop efficient high-order algorithms based on low-order collocation in analogy with similar codes based on low-order finite difference methods; some specific such procedures are proposed.This research was supported in part by the United States Office of Naval Research under Contract N00014-67-A-0126-0015.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

A congruence involving products of q-binomial coefficients

In this paper we establish a q-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime.     

On some new identities for the Fibonomial coefficients

Let F _n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.     

Modular periodicity of binomial coefficients

We prove that if the signed binomial coefficient viewed modulo p is a periodic function of i with period h in the range 0?i?k, then k+1 is a power of p, provided h is not too large compared to k. (In particular, 2h?k suffices). As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H<G, and such that 1-α∈G for all α∈G?H, then G∪{0} is a subfield.     

A new characterization of euler's gamma function by a functional equation

This paper gives a new characterization of Euler's gamma function from the aspect of complex analysis. To this end the Gauss multiplication formula is used.