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The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608101.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608103.png" /> which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608104.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608105.png" /> is defined by
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$#C+1 = 13 : ~/encyclopedia/old_files/data/L060/L.0600810 Lommel polynomial
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if TeX found to be correct.
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608106.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608107.png" /></td> </tr></table>
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The polynomial  $  R _ {m, \nu }  ( z) $
 +
of degree  $  m $
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in  $  z  ^ {-} 1 $
 +
which for  $  m = 0 , 1 ,\dots $
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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
 
or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608108.png" /></td> </tr></table>
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$$
 +
R _ {m , \nu }  ( z)  =
 +
\frac{\Gamma ( \nu + m ) }{\Gamma ( \nu ) }
 +
 
 +
\left (
 +
\frac{2}{z}
 +
\right )  ^ {m} \times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l0608109.png" /></td> </tr></table>
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$$
 +
\times
 +
{} _ {2} F _ {3} \left ( 1-
 +
\frac{m}{2}
 +
, -
 +
\frac{m}{2}
 +
; \nu , - m , 1
 +
- \nu - m ; - z  ^ {2} \right ) .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l06081010.png" /> is the Bessel function (cf. [[Bessel functions|Bessel functions]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l06081011.png" /> is the [[Hypergeometric series|hypergeometric series]]. The Lommel polynomials satisfy the relations
+
Here $  J _  \mu  ( z) $
 +
is the Bessel function (cf. [[Bessel functions|Bessel functions]]) and $  {} _ {2} F _ {3} $
 +
is the [[Hypergeometric series|hypergeometric series]]. The Lommel polynomials satisfy the relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l06081012.png" /></td> </tr></table>
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$$
 +
J _ {\nu + m }  ( z)  = J _  \nu  ( z) R _ {m , \nu }  ( z) - J _ {\nu
 +
- 1 }  ( z) R _ {m- 1 , \nu + 1 }  ( z) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060810/l06081013.png" /></td> </tr></table>
+
$$
 +
R _ {0 , \nu }  ( z)  = 1 ,\  m = 1 , 2 ,\dots .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Magnus,  F. Oberhettinger,  R.P. Soni,  "Formulas and theorems for the special functions of mathematical physics" , Springer  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Magnus,  F. Oberhettinger,  R.P. Soni,  "Formulas and theorems for the special functions of mathematical physics" , Springer  (1966)</TD></TR></table>

Latest revision as of 04:11, 6 June 2020


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

[1] W. Magnus, F. Oberhettinger, R.P. Soni, "Formulas and theorems for the special functions of mathematical physics" , Springer (1966)
How to Cite This Entry:
Lommel polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lommel_polynomial&oldid=47712
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article