Abel's Convolution Formulae through Taylor Polynomials

Year 2023, , 47 - 51, 30.11.2023

Wenchang Chu

https://doi.org/10.47087/mjm.1314434

Abstract

By making use of the Taylor polynomials, new proofs are presented
for three binomial identities including Abel’s convolution formula.

Keywords

The binomial theorem;, Taylor polynomial, Abel’s convolution formulae

References

• N. H. Abel, Beweis eines Ausdrucks, von welchem die Binomial–Formel ein einzelner Fall ist, J. Reine Angew. Math. 1 (1826), 159–160.
• W. Chu, Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Math. Appl. 283 (1994), 31–57.
• W. Chu, Generating functions and combinatorial identities, Glas. Mat. 33 (1998), 1–12.
• W. Chu, Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe, Electron. J. Combin. 17 (2010), N24.
• W. Chu, Finite differences and terminating hypergeometric series, Bull. Irish Math. Soc. 78 (2016), 31–45.
• W. Chu and L. C. Hsu, Some new applications of Gould-Hsu inversions, J. Combin. Inf. Syst. Sci. 14:1 (1990), 1–4.
• L. Comtet, Advanced Combinatorics, Dordrecht–Holland, The Netherlands, 1974.
• G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Translated from the Russian by H. H. McFadden: Translations of Mathematical Monographs 59; American Mathematical Society, Providence, RI, 1984. 286pp.
• H. W. Gould, Some generalizations of Vandermonde’s convolution, Amer. Math. Monthly 63:1 (1956), 84–91.
• H. W. Gould, Generalization of a theorem of Jensen concerning convolutions, Duke Math. J. 27 (1960), 71–76.
• H. W. Gould, Combinatorial Identities: a standardized set of tables listing 500 binomial coefficient summations, West Virginia University, Morgantown, 1972.
• H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891.
• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics (2nd edition), AddisonWesley Publ. Company, Reading, Massachusetts, 1994.
• J. L. W. V. Jensen, Sur une identite d Abel et sur d’autres formules analogues, Acta Math. 26 (1902), 307–318.
• S. G. Mohanty, Lattice Path Counting and Applications, Z. W. Birnbaum and E. Lukacs, 1979.
• T. V. Narayana, Lattice Path Combinatorics with Statistical Applications, University of Toronto Press, Toronto - 1979.
• J. Riordan, Combinatorial Identities, John Wiley Sons, Inc. New York - 1968.
• R. Sprugnoli, Riordan arrays and the Abel–Gould identity, Discrete Math. 142 (1995), 213– 233.
• V. Strehl, Identities of Rothe–Abel–Schl¨afli–Hurwitz–type, Discrete Math. 99:1–3 (1992), 321– 340.
• P. J. Zucker, Problem 1578, Math. Mag. 72:1 (1999), page 237; Solution: ibid 73:3 (2000), 243–245.