Extended Bell and Stirling numbers from hypergeometric exponentiation (Q2736386)
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scientific article; zbMATH DE number 1638689
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Extended Bell and Stirling numbers from hypergeometric exponentiation |
scientific article; zbMATH DE number 1638689 |
Statements
29 August 2001
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extended Bell numbers
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Stirling numbers
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recurrence relations
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Extended Bell and Stirling numbers from hypergeometric exponentiation (English)
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Extended Bell numbers \(b_L(n)\) are defined by the relation NEWLINE\[NEWLINEe^{[_0F_L(z)-1]}=\sum_{n=0}^\infty b_L(n)\frac{z^n}{(n!)^{L+1}}NEWLINE\]NEWLINE where \({}_0F_L(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^{L+1}}\). The authors find recurrence relations for \(b_L(n)\), and also for Stirling type numbers \(S_L(n,l)\) defined by NEWLINE\[NEWLINE \frac{\left( {}_0F_L(z)-1\right)^l}{l!}=\sum_{n=l}^\infty \frac{S_L(n,l)}{(n!)^{L+1}}z^n,\quad L=0,1,2,\ldots.NEWLINE\]
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