ACTA issues

On analytical and diophantine properties of a family of counting polynomials

Peter Kirschenhofer, Attila Pethő, Robert F. Tichy

Acta Sci. Math. (Szeged) 65:1-2(1999), 47-59

Abstract. We study the polynomials $p_n(x)$ where $p_n(k)$ counts the number of integer-coordinate lattice points $(x_1,\ldots,x_n)$ with $\sum_{i=1}^n|x_i|\le k $. Using the fact that the polynomials $i^nn!p_n(-1/2-ix/2)$ are the classical Meixner polynomials of the second kind, we are able to prove finiteness results on the number of solutions of the diophantine equation $p_n(x)=p_m(y)$.

AMS Subject Classification (1991): 11D41, 26C10, 33C25, 05A15

Received October 20, 1997 and in revised form October 7, 1998. (Registered under 2671/2009.)