# 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

- Harry Bateman:
*Some Properties of a certain Set of Polynomials*(1933)... (previous)... (next) $3.$ - 1960: Earl David Rainville:
*Special Functions*... (previous) ... (next): $148. (1)$

**Orthogonal polynomials**