Double Fun with Double Factorials

SummaryThe double factorial of n may be defined inductively by (n + 2)!! = (n + 2)(n)!! with (0)!! = (1)!! = 1. Alternatively we may define this notion by the two relations (2n)!! = 2 ·4 · 6 · 8…(2n) =2nn! and (2n - 1)!! = 1 · 3 · 5 · 7…(2n - 1) = (2n)!/2n!. Our object is to exhibit some properties and identities for the double factorials. Furthermore, we extend the notion of double factorial to the binomial coefficients by introducing double factorial binomial coefficients. The double factorial binomial coefficient is defined as We derive identities and generating functions involving these double factorial binomial coefficients.