Bernoulli numbers and confluent hypergeometric functions
by
Karl Dilcher
Dalhousie University

If the reciprocal of the confluent hypergeometric function M(s+1;r+s+2;z) is taken as an exponential generating function, the resulting sequences of numbers have many properties resembling those of the Bernoulli numbers, which arise when r=s=0.

Other special cases include van der Pol numbers and generalized van der Pol numbers (for positive integers r=s), as well as various sequences of combinatorial numbers and linear recurrence sequences of arbitrary orders. Explicit expressions resembling Euler's formula involve sums of powers of the zeros of the corresponding confluent hypergeometric functions.

Date received: March 14, 2000