Number Theoretic Applications of Polynomials with Rational Coefficients Defined by Extremality Conditions

Abstract

It is well known that classes of polynomials in one variable defined by various extremality conditions play an extremely important role in complex analysis. Among these classes we find orthogonal polynomials (especially classical orthogonal polynomials expressed as hypergeometric polynomials) and polynomials least deviating from zero on a given continuum (Chebicheff polynomials). Orthogonal polynomials of the first and second kind appear as denominators and numerators of the Padé approximations to functions of classical analysis and satisfy familiar three-term linear recurrences. These polynomials were used repeatedly to study diophantine approximations of values of functions of classical analysis, especially exponential and logarithmic functions at rational points x = p/q [1], [2], [3], [4], [5]. The methods of Padé approximation in diophantine approximations are quite powerful and convenient to use, since they replace the problem of rational approximations to numbers with the approximations of functions. There are, however, arithmetic restrictions on rational approximations to functions if they are to be used for diophantine approximations. The main restriction on polynomials here is to have rational integer coefficients or rational coefficients with a controllable denominator. Such arithmetic restrictions transform a typical problem of classical analysis into an unusual mixture of arithmetic and analytic difficulties. For example, recurrences defining orthogonal polynomials must be of a special type to guarantee that their solutions will have hounded denominators. In this paper we consider various classes of polynomials generated by imposing arithmetic restrictions on classical approximation theory problems (orthogonal or Chebicheff polynomials).