oronto Function

The function defined by

(1)

(Heatley 1943; Abramowitz and Stegun 1972, p. 509), where  is a confluent hypergeometric function of the first kind and  is the gamma function.

Heatley originally defined the function in terms of the integral

(2)

where  is a modified Bessel function of the first kind, which is similar to an integral of Watson (1966, p. 394), with Watson's  changed to  and a few other minor changes of variables. In terms of this function,

(3)

(Heatley 1943). Heatley (1943) also gives a number of recurrences and other identities satisfied by .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 509, 1972.

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 268, 1981.

Heatley, A. H. "A Short Table of the Toronto Function." Trans. Roy. Soc. Canada 37, 13-29, 1943.

Slater, L. J. Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, p. 99, 1960.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.