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= | ||
Revision as of 11:47, 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
References
- Harry Bateman: Some Properties of a certain Set of Polynomials (1933)... (previous)... (next) $3.$
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $148. (1)$
