ASYMPTOTIC EXPANSIONS OF INTEGRAL MEAN OF POLYGAMMA FUNCTIONS

Let s,t be two given real numbers, s not equal t and m is an element of N. We determine the coefficients a(j)(s,t) in the asymptotic expansion of integral (or differential) mean of polygamma functions Psi((m))(x) : 1/t - s integral(t)(s)Psi((m))(x + u) du similar to psi((m)) (x Sigma(infinity)(j=0) a(j)(s,t)/x(j)), x -> infinity. We derive the recursive relations for polynomials a(j)(t, s), and also as polynomials in intrinsic variables alpha = 1/2 (s + t - 1), beta = 1/4 [1 - (t - s)(2)]. We derive also the main properties of these polynomials and as a consequence the asymptotic formula for shifted variables.