The Lommel polynomials arise from the equation
|
(1)
|
where
is a Bessel function of the first kind
and
is a complex number (Watson 1966, p. 294).
The function is given by
|
(2)
|
(Watson 1966, §9.61, p. 297, eqn. 5; Erdelyi et al. 1981, §7.5.2, p. 34, eqn. 25), where is a generalized
hypergeometric function and is a gamma function,
and
|
(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
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.Referenced on Wolfram|Alpha
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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