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Toronto Function

The function defined by

T(m,n,r)=r^(2n-m+1)e^(-r^2)(Gamma(1/2m+1/2))/(n!)_1F_1(1/2(m+1);n+1;r^2)
(1)

(Heatley 1943; Abramowitz and Stegun 1972, p. 509), where _1F_1(a;b;z) is a confluent hypergeometric function of the first kind and Gamma(z) is the gamma function.

Heatley originally defined the function in terms of the integral

T(m,n,p,a)=int_0^inftyt^(-n)e^(-p^2t^2)I_n(2at)dt,
(2)

where I_n(x) is a modified Bessel function of the first kind, which is similar to an integral of Watson (1966, p. 394), with Watson's J_nu(at) changed to I_n(2at) and a few other minor changes of variables. In terms of this function,

T(m,n,r)=2r^(n-m+1)e^(-r^2)T(m,n,1,r)
(3)

(Heatley 1943). Heatley (1943) also gives a number of recurrences and other identities satisfied by T(m,n,r).

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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.

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Toronto Function

Cite this as:

Weisstein, Eric W. "Toronto Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TorontoFunction.html

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