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Related to [[Bessel functions|Bessel functions]], [[#References|[a2]]], the Bessel polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b1104101.png" /> satisfy
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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/b/b110/b110410/b1104102.png" /></td> </tr></table>
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Related to [[Bessel functions|Bessel functions]], [[#References|[a2]]], the Bessel polynomials  $  \{ y _ {n} ( x,a,b ) \} _ {n = 0 }  ^  \infty  $
 +
satisfy
 +
 
 +
$$
 +
x  ^ {2} y ^ {\prime \prime } + ( ax + b ) y _ {n}  ^  \prime  - n ( n + a - 1 ) y = 0
 +
$$
  
 
and are given by
 
and are given by
  
<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/b/b110/b110410/b1104103.png" /></td> </tr></table>
+
$$
 +
y _ {n} ( x,a,b ) = \sum _ {k = 0 } ^ { n }  {
 +
\frac{n!  \Gamma ( n + k + a - 1 ) ( {x / b } )  ^ {k} }{k! ( n - k ) !  \Gamma ( n + a - 1 ) }
 +
} .
 +
$$
  
The ordinary Bessel polynomials are those found with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b1104104.png" />, [[#References|[a2]]].
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The ordinary Bessel polynomials are those found with $  a = b = 2 $,  
 +
[[#References|[a2]]].
  
 
The moments associated with the Bessel polynomials satisfy
 
The moments associated with the Bessel polynomials satisfy
  
<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/b/b110/b110410/b1104105.png" /></td> </tr></table>
+
$$
 +
( n + a - 1 ) \mu _ {n} + b \mu _ {n - 1 }  = 0, \quad n =0,1 \dots
 +
$$
  
and are given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b1104106.png" />.
+
and are given by $  \mu _ {n} = { {( - b ) ^ {n + 1 } } / {a ( a + 1 ) \dots ( a + n - 1 ) } } $.
  
 
The weight equation is
 
The weight equation is
  
<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/b/b110/b110410/b1104107.png" /></td> </tr></table>
+
$$
 +
x  ^ {2} w  ^  \prime  + ( ( 2 - a ) x - b ) w = N ( x ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b1104108.png" /> is any function with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b1104109.png" /> moments. This equation has been solved when
+
where $  N ( x ) $
 +
is any function with 0 $
 +
moments. This equation has been solved when
  
<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/b/b110/b110410/b11041010.png" /></td> </tr></table>
+
$$
 +
N ( x ) = H ( x ) e ^ {- x ^ { {1 / 4 } } } { \mathop{\rm sn} } x ^ { {1 / 4 } } ,
 +
$$
  
 
where
 
where
  
<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/b/b110/b110410/b11041011.png" /></td> </tr></table>
+
$$
 +
H ( x ) = \left \{
 +
\begin{array}{l}
 +
{1, \  x \geq  0, } \\
 +
{0, \  x < 0, }
 +
\end{array}
 +
\right .
 +
$$
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041012.png" /> (no restriction), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041014.png" />, [[#References|[a3]]]. The weight for the ordinary Bessel polynomials was found by S.S. Kim, K.H. Kwon and S.S. Han, [[#References|[a1]]], after over 40 years of search.
+
when $  b = 2 $(
 +
no restriction), $  a - 2 = 2 \alpha $
 +
and  $  \alpha > 6 ( {2 / \pi } )  ^ {4} $,  
 +
[[#References|[a3]]]. The weight for the ordinary Bessel polynomials was found by S.S. Kim, K.H. Kwon and S.S. Han, [[#References|[a1]]], after over 40 years of search.
  
 
Using the three-term recurrence relation
 
Using the three-term recurrence relation
  
<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/b/b110/b110410/b11041015.png" /></td> </tr></table>
+
$$
 +
( n + a - 1 ) ( 2n + a - 2 ) y _ {n + 1 }  ( x,a,b ) =
 +
$$
  
<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/b/b110/b110410/b11041016.png" /></td> </tr></table>
+
$$
 +
=  
 +
\left [ ( 2n + a ) ( 2n + a - 2 ) \left ( {
 +
\frac{x}{b}
 +
} \right ) + ( a - 2 ) \right ]  \cdot
 +
$$
  
<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/b/b110/b110410/b11041017.png" /></td> </tr></table>
+
$$
 +
\cdot
 +
( 2n + a - 1 ) y _ {n} + n ( 2n + a ) y _ {n - 1 }  ,
 +
$$
  
the norm square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041018.png" /> is easily calculated and equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041019.png" />, [[#References|[a2]]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041020.png" />. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041021.png" /> generates a [[Krein space|Krein space]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110410/b11041022.png" />.
+
the norm square $  \int _ {0}  ^  \infty  {g _ {n} ( x,a,b )  ^ {2} w ( x ) }  {dx } $
 +
is easily calculated and equals $  { {( - b ) ^ {k + 1 } k ^ {( n ) } } / {( k + a + n _ {1} ) ^ {( k + n ) } } } $,  
 +
[[#References|[a2]]], where $  x ^ {( k ) } = x ( x - 1 ) \dots ( x - k + 1 ) $.  
 +
Clearly, $  w $
 +
generates a [[Krein space|Krein space]] on $  [ 0, \infty ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.S. Kim,  K.H. Kwon,  S.S. Han,  "Orthogonalizing weights of Tchebychev sets of polynomials"  ''Bull. London Math. Soc.'' , '''24'''  (1992)  pp. 361–367</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.L. Krall,  O. Frink,  "A new class of orthogonal polynomials: The Bessel polynomials"  ''Trans. Amer. Math. Soc.'' , '''63'''  (1949)  pp. 100–115</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Maroni,  "An integral representation for the Bessel form"  ''J. Comp. Appl. Math.'' , '''57'''  (1995)  pp. 251–260</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.S. Kim,  K.H. Kwon,  S.S. Han,  "Orthogonalizing weights of Tchebychev sets of polynomials"  ''Bull. London Math. Soc.'' , '''24'''  (1992)  pp. 361–367</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.L. Krall,  O. Frink,  "A new class of orthogonal polynomials: The Bessel polynomials"  ''Trans. Amer. Math. Soc.'' , '''63'''  (1949)  pp. 100–115</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Maroni,  "An integral representation for the Bessel form"  ''J. Comp. Appl. Math.'' , '''57'''  (1995)  pp. 251–260</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


Related to Bessel functions, [a2], the Bessel polynomials $ \{ y _ {n} ( x,a,b ) \} _ {n = 0 } ^ \infty $ satisfy

$$ x ^ {2} y ^ {\prime \prime } + ( ax + b ) y _ {n} ^ \prime - n ( n + a - 1 ) y = 0 $$

and are given by

$$ y _ {n} ( x,a,b ) = \sum _ {k = 0 } ^ { n } { \frac{n! \Gamma ( n + k + a - 1 ) ( {x / b } ) ^ {k} }{k! ( n - k ) ! \Gamma ( n + a - 1 ) } } . $$

The ordinary Bessel polynomials are those found with $ a = b = 2 $, [a2].

The moments associated with the Bessel polynomials satisfy

$$ ( n + a - 1 ) \mu _ {n} + b \mu _ {n - 1 } = 0, \quad n =0,1 \dots $$

and are given by $ \mu _ {n} = { {( - b ) ^ {n + 1 } } / {a ( a + 1 ) \dots ( a + n - 1 ) } } $.

The weight equation is

$$ x ^ {2} w ^ \prime + ( ( 2 - a ) x - b ) w = N ( x ) , $$

where $ N ( x ) $ is any function with $ 0 $ moments. This equation has been solved when

$$ N ( x ) = H ( x ) e ^ {- x ^ { {1 / 4 } } } { \mathop{\rm sn} } x ^ { {1 / 4 } } , $$

where

$$ H ( x ) = \left \{ \begin{array}{l} {1, \ x \geq 0, } \\ {0, \ x < 0, } \end{array} \right . $$

when $ b = 2 $( no restriction), $ a - 2 = 2 \alpha $ and $ \alpha > 6 ( {2 / \pi } ) ^ {4} $, [a3]. The weight for the ordinary Bessel polynomials was found by S.S. Kim, K.H. Kwon and S.S. Han, [a1], after over 40 years of search.

Using the three-term recurrence relation

$$ ( n + a - 1 ) ( 2n + a - 2 ) y _ {n + 1 } ( x,a,b ) = $$

$$ = \left [ ( 2n + a ) ( 2n + a - 2 ) \left ( { \frac{x}{b} } \right ) + ( a - 2 ) \right ] \cdot $$

$$ \cdot ( 2n + a - 1 ) y _ {n} + n ( 2n + a ) y _ {n - 1 } , $$

the norm square $ \int _ {0} ^ \infty {g _ {n} ( x,a,b ) ^ {2} w ( x ) } {dx } $ is easily calculated and equals $ { {( - b ) ^ {k + 1 } k ^ {( n ) } } / {( k + a + n _ {1} ) ^ {( k + n ) } } } $, [a2], where $ x ^ {( k ) } = x ( x - 1 ) \dots ( x - k + 1 ) $. Clearly, $ w $ generates a Krein space on $ [ 0, \infty ) $.

References

[a1] S.S. Kim, K.H. Kwon, S.S. Han, "Orthogonalizing weights of Tchebychev sets of polynomials" Bull. London Math. Soc. , 24 (1992) pp. 361–367
[a2] H.L. Krall, O. Frink, "A new class of orthogonal polynomials: The Bessel polynomials" Trans. Amer. Math. Soc. , 63 (1949) pp. 100–115
[a3] P. Maroni, "An integral representation for the Bessel form" J. Comp. Appl. Math. , 57 (1995) pp. 251–260
How to Cite This Entry:
Bessel polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_polynomials&oldid=46035
This article was adapted from an original article by A.M. Krall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article