On analytical and diophantine properties of a family of counting polynomials. (English) Zbl 0983.11013

Kirschenhofer, Peter; Pethő, Attila; Tichy, Robert F.

Acta Sci. Math. 65, No. 1-2, 47-59 (1999).

Let \(p_n(k)\) be the number of lattice points \((x_1,\dots,x_n)\in {\mathbb Z}^n\) such that \(\sum^n_{i=1}|x_i|\leq k\). In this paper the authors first prove that the polynomials \(i^n n!p_n(-(1+ix)/2)\) are the classical Meixner polynomials of the second kind. Using this fact, an explicit bound for the prime number function \(\pi(x)\) and the famous theorem of A. Baker about the integer solutions of the hyperelliptic equation, they show that for \(n=2,4\) and \(m\geq 3\) the equation \(p_n(x)=p_m(y)\) has only finitely many integer solutions in \(x\), \(y\) which are effectively computable. Moreover, using a well known result of Davenport, Lewis and Schinzel about the integer solutions of the equations \(f(x)=g(x)\) (where \(f(x)\), \(g(x)\) are polynomials with integer coefficients), they prove that the equation \(p_n(x)=p_m(y)\) has only finitely integer solutions in \(x\), \(y\), provided that \(m,n\geq 2\) with \(m\not\equiv n \pmod 2\) as well as that \(2\leq n< m\leq 103\).

Reviewer: Dimitrios Poulakis (Thessaloniki)

MSC:

11D45	Counting solutions of Diophantine equations
11D41	Higher degree equations; Fermat’s equation
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05A15	Exact enumeration problems, generating functions
26C10	Real polynomials: location of zeros

Keywords:

combinatorial Diophantine equations; Meixner polynomials; Sheffer sequence