Difference between revisions of "Bateman F"

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The Bateman polynomials $Z_n$ are defined by the formula
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The Bateman polynomials $F_n$ are defined for $n=0,1,2,\ldots$ by the formula
$$Z_n(x) = {}_2F_2(-n,n+1;1;1;x),$$
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$$F_n(z) = {}_3F_2 \left( -n, n+1, \dfrac{z+1}{2}; 1,1;1 \right),$$
where ${}_2F_2$ is a [[generalized hypergeometric function]].
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where ${}_3F_2$ denotes the [[generalized hypergeometric function]]. The first few Bateman polynomials are
 +
$$\begin{array}{l|l}
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n & F_n(z) \\
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\hline
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0 & 1 \\
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1 & -z \\
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2 & \dfrac{3}{4}z^2+\dfrac{1}{4} \\
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3 & -\dfrac{5}{12}z^3-\dfrac{7}{12}z^2 \\
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4 & \dfrac{35}{192}z^4 + \dfrac{65}{96}z^2+\dfrac{9}{64} \\
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5 & -\dfrac{21}{320}z^5 - \dfrac{49}{96}z^3 - \dfrac{407}{960}z \\
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\vdots & \vdots
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\end{array}$$
  
 
=Properties=
 
=Properties=
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[[Generating relation for Bateman F]]<br />
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[[Three-term recurrence for Bateman F]]<br />
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[[Orthogonality of Bateman F on R]]<br />
  
 
=References=
 
=References=
* {{PaperReference|Some Properties of a certain Set of Polynomials|1933|Harry Bateman}} $3.$
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* {{PaperReference|Some Properties of a certain Set of Polynomials|1933|Harry Bateman|prev=findme|next=findme}} $3.$
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* {{PaperReference|The Polynomial Fn(x)|1934|Harry Bateman}}
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* {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=Generating relation for Bateman F}}: $148. (1)$
  
 
{{:Orthogonal polynomials footer}}
 
{{:Orthogonal polynomials footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

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

Orthogonal polynomials