Lommel polynomial

The polynomial $ R _ {m, \nu } ( z) $ of degree $ m $ in $ z ^ {-} 1 $ which for $ m = 0 , 1 ,\dots $ and any $ \nu $ is defined by

$$ R _ {m , \nu } ( z) = $$

$$ = \ \frac{\pi z }{2 \sin \nu \pi } [ J _ {\nu + m } ( z) J _ { - \nu + 1 } ( z) + (- 1) ^ {m} J _ {- \nu - m } ( z) J _ {\nu - 1 } ( z)] $$

or

$$ R _ {m , \nu } ( z) = \frac{\Gamma ( \nu + m ) }{\Gamma ( \nu ) } \left ( \frac{2}{z} \right ) ^ {m} \times $$

$$ \times {} _ {2} F _ {3} \left ( 1- \frac{m}{2} , - \frac{m}{2} ; \nu , - m , 1 - \nu - m ; - z ^ {2} \right ) . $$

Here $ J _ \mu ( z) $ is the Bessel function (cf. Bessel functions) and $ {} _ {2} F _ {3} $ is the hypergeometric series. The Lommel polynomials satisfy the relations

$$ J _ {\nu + m } ( z) = J _ \nu ( z) R _ {m , \nu } ( z) - J _ {\nu - 1 } ( z) R _ {m- 1 , \nu + 1 } ( z) , $$

$$ R _ {0 , \nu } ( z) = 1 ,\ m = 1 , 2 ,\dots . $$

References[edit]