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

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 \(2\leq n< m\leq 103\).

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
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