Difference between revisions of "Bateman F"

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The Bateman polynomials are [[orthogonal polynomials]] defined by
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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
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$$\begin{array}{l|l}
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n & F_n(z) \\
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\hline
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0 & 1 \\
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1 & -z \\
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2 & \dfrac{3}{4}z^2+\dfrac{1}{4} \\
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3 & -\dfrac{5}{12}z^3-\dfrac{7}{12}z^2 \\
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4 & \dfrac{35}{192}z^4 + \dfrac{65}{96}z^2+\dfrac{9}{64} \\
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5 & -\dfrac{21}{320}z^5 - \dfrac{49}{96}z^3 - \dfrac{407}{960}z \\
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\vdots & \vdots
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\end{array}$$
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Generating relation for Bateman F]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[Three-term recurrence for Bateman F]]<br />
$$n^2(2n-3)Z_n(x)-(2n-1)[3n^2-6n+2-2(2n-3)x]Z_{n-1}(x)+(2n-3)[3n^2-6n+2+2(2n-1)x]Z_{n-2}(x)-(2n-1)(n-2)^2Z_{n-3}(x)=0.$$
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[[Orthogonality of Bateman F on R]]<br />
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$xZ_n'(x)-nZ_n(x)=-nZ_{n-1}(x)-xZ_{n-1}'(x).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$xZ_n'(x)-nZ_n(x)=-\displaystyle\sum_{k=0}^{n-1} Z_k(x) - 2x\displaystyle\sum_{k=0}^{n-1} Z_k'(x).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$xZ_n'(x)-nZ_n(x)=\displaystyle\sum_{k=0}^{n-1} (-1)^{n-k}(2k+1)Z_k(x).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$\dfrac{1}{1-t} {}_1F_1 \left( \dfrac{1}{2} ; 1 ; -\dfrac{4xt}{(1-t)^2} \right) = \displaystyle\sum_{k=0}^{\infty} Z_n(x)t^n,$$
 
where ${}_1F_1$ denotes the [[generalized hypergeometric function]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
  
 
=References=
 
=References=
Rainville "Special Functions"
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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}}
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[[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