Gould polynomials
From HandWiki
In mathematics the Gould polynomials Gn(x; a,b) are polynomials introduced by H. W. Gould and named by Roman in 1984.[1] They are given by [2]
- [math]\displaystyle{ \displaystyle \exp(x f^{-1}(t)) = \sum_{n=0}^{\infty} G_n(x;a,b)\frac{t^n}{n!} }[/math]
where
- [math]\displaystyle{ f(t)=e^{at}(e^{bt}-1) }[/math] so [math]\displaystyle{ f^{-1}(t)=\frac{1}{b}\sum_{k=1}^{\infty}\binom{-(b+ak)/b}{k-1}\frac{t^k}{k} }[/math]
References
- ↑ Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, 111, London: Academic Press Inc. Harcourt Brace Jovanovich Publishers, Reprinted by Dover, 2005, ISBN 978-0-12-594380-2, https://books.google.com/books?id=JpHjkhFLfpgC
- ↑ Gould, H. W. (1961), "A series transformation for finding convolution identities", Duke Math. J. Volume 28, Number 2, 193-202.
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