A GENERALIZED EXPONENTIAL FUNCTION

Russell Euler

Northwest Missouri State University

Let p be a positive integer. Using the ratio test, it can be shown that

(1)  S(p) = ∑_{n=0}^{∞} \frac{x^{n}}{(pn)!}

converges for |x| < ∞. This follows from the fact

lim_{n→∞} \frac{(pn)!}{(p(n + 1))!} = 0.

The purpose of this paper is to express (1) in terms of a hypergeometric function. The special functions that will be used are reviewed first.

The factorial function is defined by

(a)_{n} = a(a + 1) ⋯ (a + n − 1)

for any a and for n ≥ 1 and, (a)_{0} = 1, for a ≠ 0. In particular, n! = (1)_{n} and, from page 9 of [1],

(a)_{nk} = k^{nk} ∏_{i=1}^{k} (\frac{a + i − 1}{k})_{n} .

So,

(pn)! = (1)_{pn}

= p^{pn} (\frac{1}{p})_{n} (\frac{2}{p})_{n} ⋯ (\frac{p − 1}{p})_{n} (\frac{p}{p})_{n}

= p^{pn} (\frac{1}{p})_{n} (\frac{2}{p})_{n} ⋯ (\frac{p − 1}{p})_{n} (1)_{n}

= p^{pn} (\frac{1}{p})_{n} (\frac{2}{p})_{n} ⋯ (\frac{p − 1}{p})_{n} n! . (2)

The generalized hypergeometric function is defined by

pF_{q}[a_{1}, …, a_{p}; b_{1}, …, b_{q}; x] = 1 + ∑_{n=1}^{∞} \frac{(a_{1})_{n} ⋯ (a_{p})_{n}}{(b_{1})_{n} ⋯ (b_{q})_{n}} · \frac{x^{n}}{n!}

for all b_{i} ≠ 0, −1, −2, . . .. It is known that this series diverges for all x ≠ 0 if p > q + 1; converges for all x if p ≤ q; and converges for |x| < 1 if p = q + 1. These facts can be established by using the ratio test.

Substituting identity (2) into (1) gives

S(p) = ∑_{n=0}^{∞} \frac{x^{n}}{p^{pn} (\frac{1}{p})_{n} (\frac{2}{p})_{n} ⋯ (\frac{p−1}{p})_{n} n!}

= ∑_{n=0}^{∞} \frac{1}{(\frac{1}{p})_{n} (\frac{2}{p})_{n} ⋯ (\frac{p−1}{p})_{n}} · \frac{(x/p^{p})^{n}}{n!}

=_{0}F_{p−1} (−; \frac{1}{p}, \frac{2}{p}, ⋯, \frac{p − 1}{p}; \frac{x}{p^{p}} ) . (3)

In particular, if p = 1, then (1) is the series representation for e^{x} and (3) becomes

S(1) =_{0}F_{0}(−; −; x) = e^{x}.

This agrees with the identity e^{z} =_{0}F_{0}(z) given on page 209 of [1].

\underline{Reference}

1. Y. Luke, The Special Functions and Their Approximations, Vol. 1, Academic Press, 1969.