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Lommel Polynomial

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The Lommel polynomials R_(m,nu)(z) arise from the equation

J_(m+nu)(z)=J_nu(z)R_(m,nu)(z)-J_(nu-1)(z)R_(m-1,nu+1)(z),
(1)

where J_nu(z) is a Bessel function of the first kind and nu is a complex number (Watson 1966, p. 294). The function is given by

R_(m,nu)(z)=(Gamma(nu+m))/(Gamma(nu)(z/2)^m)_2F_3(1/2(1-m),-1/2m;nu,-m,1-nu-m;-z^2)
(2)

(Watson 1966, §9.61, p. 297, eqn. 5; Erdelyi et al. 1981, §7.5.2, p. 34, eqn. 25), where _2F_3(a,b;c,d,e;z) is a generalized hypergeometric function and Gamma(z) is a gamma function, and

R_(m,nu)(z)=(piz)/(2sin(nupi))[J_(nu+m)(z)J_(-nu+1)(z)+(-1)^mJ_(-nu-m)(z)J_(nu-1)(z)]
(3)

(Watson 1966, §9.61, p. 295, eqn. 2; Erdelyi et al. 1981, §7.5.2, pp. 34-35, eqn. 26).

Since (1) must reduce to the usual recurrence formula for Bessel functions, it follows that

R_(0,nu)(z)=1
(4)
R_(1,nu)(z)=(2nu)/z.
(5)

See also

Lommel Differential Equation, Lommel Function

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References

Erdelyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Lommel's Polynomials." §7.5.2 in Higher Transcendental Functions, Vol. 2. Krieger, pp. 34-35, 1981.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980.Watson, G. N. §9.6-9.65 in A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 294-303, 1966.

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Lommel Polynomial

Cite this as:

Weisstein, Eric W. "Lommel Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LommelPolynomial.html

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