## 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