Bessel polynomials

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