How To Generalize (and Not To Generalize) the Chu–Vandermonde Identity

Abstract

We consider two different interpretations of the celebrated Chu–Vandermonde identity: as an identity for polynomials, and as an identity for infinite matrices. Each interpretation leads to a class of possible generalizations, and in both cases we obtain a complete characterization of the solutions.

• MSC: Primary 05A19
• Secondary 13B25
• 15A24

Acknowledgments

I wish to thank Mathias Pétréolle and Bao-Xuan Zhu for helpful conversations, and the referees for helpful suggestions. This research was supported by the UK Engineering and Physical Sciences Research Council under Grant EP/N025636/1.

Notes

1 As part of the definition of “sequence of binomial type,” Rota et al. [15, 21] and many subsequent authors [7, 8, 19, 20] imposed the additional condition that $\mathrm{deg}{F}_n=n$ exactly. But this condition is irrelevant for our purposes, so we refrain from imposing it.

2 As part of the definition of “Sheffer sequence,” some authors (e.g., [19, p. 2]) impose the additional conditions $A(0)\ne 0$, $B(0)=0$, and $B\prime (0)\ne 0$. But these conditions are irrelevant for our purposes, so we refrain from imposing them.

3 References [2, 17] call this a “generalized Riordan array,” but we prefer to avoid this term because it has already been used, in a highly cited paper [29], for a completely unrelated generalization of Riordan arrays.

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Notes on contributors

Alan D. Sokal

ALAN SOKAL spent his first two decades as a mathematical physicist trying desperately to avoid combinatorics. Having subsequently seen the light, he now works principally on problems at the boundary between combinatorics and analysis. He also sometimes moonlights as an amateur philosopher of science. He is Professor of Mathematics at University College London and Professor Emeritus of Physics at New York University.