# Difference between revisions of "Bateman F"

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* {{PaperReference|Some Properties of a certain Set of Polynomials|1933|Harry Bateman|prev=findme|next=findme}} $3.$ | * {{PaperReference|Some Properties of a certain Set of Polynomials|1933|Harry Bateman|prev=findme|next=findme}} $3.$ | ||

+ | * {{PaperReference|The Polynomial Fn(x)|1934|Harry Bateman}} | ||

* {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=Generating relation for Bateman F}}: $148. (1)$ | * {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=Generating relation for Bateman F}}: $148. (1)$ | ||

## Latest revision as of 11:57, 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) $3.$ - Harry Bateman:
*The Polynomial Fn(x)*(1934) - 1960: Earl David Rainville:
*Special Functions*... (previous) ... (next): $148. (1)$

**Orthogonal polynomials**