Another "beta function" defined in terms of an integral is the "exponential" beta function, given by β_n(z) | congruent | integral_(-1)^1 t^n e^(-z t) d t | = | n!z^(-(n + 1))[e^z sum_(k = 0)^n ((-1)^k z^k)/(k!) - e^(-z) sum_(k = 0)^n z^k/(k!)]. If n is an integer, then β_n(z) = (-1)^(n + 1) E_(-n)(-z) - E_(-n)(z), where E_n(z) is the E_n-Function. The exponential beta function satisfies the recurrence relation z β_n(z) = (-1)^n e^z - e^(-z) + n β_(n - 1)(z).
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