New Expression for CDF of $$\chi _\nu '^2(\lambda )$$ Distribution and Marcum $$Q_1$$ Function

Abstract

AbstractClosed form expression is established for the cumulative distribution function of the non-central $$\chi _\nu '^2(\lambda )$$ χ ν ′ 2 ( λ ) probability distribution with integer degrees of freedom $$\nu $$ ν . This expression includes the Humbert $$\Phi _1$$ Φ 1 (confluent Appell) hypergeometric function of two variables. A new representation of the Marcum $$Q_1$$ Q 1 function of the first order is obtained, interconnection formula accompanied with a reduction formula for specific, equal unit-parameter case of the Humbert function $$\Phi _1$$ Φ 1 and the function $$\Phi _3$$ Φ 3 are established.