Asymptotic Expansions of Integral Mean of Polygamma Functions

Let s,t be two given real numbers, s = t and m ∈ N . We determine the coefficients aj(s,t) in the asymptotic expansion of integral (or differential) mean of polygamma functions ψ (m)(x) : 1 t− s ∫ t s ψ (m)(x+u)du ∼ ψ (m) ( x ∞ ∑ j=0 aj(s,t) x j ) , x → ∞. We derive the recursive relations for polynomials aj(t,s) , and also as polynomials in intrinsic variables α = 2 (s + t − 1) , β = 4 [1− (t − s)2] . We derive also the main properties of these polynomials and as a consequence the asymptotic formula for shifted variables. Mathematics subject classification (2010): Primary 33B15; Secondary 41A60.