Difference between revisions of "Bateman F"

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The Bateman polynomials $F_n$ are defined for $n=0,1,2,\ldots$ by the formula
 
The Bateman polynomials $F_n$ are defined for $n=0,1,2,\ldots$ by the formula
 
$$F_n(z) = {}_3F_2 \left( -n, n+1, \dfrac{z+1}{2}; 1,1;1 \right),$$
 
$$F_n(z) = {}_3F_2 \left( -n, n+1, \dfrac{z+1}{2}; 1,1;1 \right),$$
where ${}_3F_2$ denotes the [[generalized hypergeometric function]].
+
where ${}_3F_2$ denotes the [[generalized hypergeometric function]]. The first few Bateman polynomials are
 +
$$\begin{array}{l|l}
 +
n & F_n(z) \\
 +
\hline
 +
0 & 1 \\
 +
1 & -z \\
 +
2 & \dfrac{3}{4}z^2+\dfrac{1}{4} \\
 +
3 & -\dfrac{5}{12}z^3-\dfrac{7}{12}z^2 \\
 +
4 & \dfrac{35}{192}z^4 + \dfrac{65}{96}z^2+\dfrac{9}{64} \\
 +
5 & -\dfrac{21}{320}z^5 - \dfrac{49}{96}z^3 - \dfrac{407}{960}z \\
 +
\vdots & \vdots
 +
\end{array}$$
  
 
=Properties=
 
=Properties=
 
[[Generating relation for Bateman F]]<br />
 
[[Generating relation for Bateman F]]<br />
 
[[Three-term recurrence for Bateman F]]<br />
 
[[Three-term recurrence for Bateman F]]<br />
 +
[[Orthogonality of Bateman F on R]]<br />
  
 
=References=
 
=References=

Revision as of 11:48, 10 October 2019

The Bateman polynomials $F_n$ are defined for $n=0,1,2,\ldots$ by the formula $$F_n(z) = {}_3F_2 \left( -n, n+1, \dfrac{z+1}{2}; 1,1;1 \right),$$ where ${}_3F_2$ denotes the generalized hypergeometric function. The first few Bateman polynomials are $$\begin{array}{l|l} n & F_n(z) \\ \hline 0 & 1 \\ 1 & -z \\ 2 & \dfrac{3}{4}z^2+\dfrac{1}{4} \\ 3 & -\dfrac{5}{12}z^3-\dfrac{7}{12}z^2 \\ 4 & \dfrac{35}{192}z^4 + \dfrac{65}{96}z^2+\dfrac{9}{64} \\ 5 & -\dfrac{21}{320}z^5 - \dfrac{49}{96}z^3 - \dfrac{407}{960}z \\ \vdots & \vdots \end{array}$$

Properties

Generating relation for Bateman F
Three-term recurrence for Bateman F
Orthogonality of Bateman F on R

References

Orthogonal polynomials