Reciprocal Formulae on Binomial Convolutions of Hagen-Rothe Type

Wenchang Chu

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 3, page 591-605
  • ISSN: 0392-4041

Abstract

top
By means of duplicate inverse series relations, we investigate dual relations of four binomial convolution identities. Four classes of reciprocal formulae on binomial convolutions of Hagen-Rothe type are established. They reflect certain “reciprocity” on the Hagen-Rothe-like convolutions in the sense that each binomial summation involved has no closed form in general, but their sum and difference in pairs do have simple expressions in a single term of binomial coefficients.

How to cite

top

Chu, Wenchang. "Reciprocal Formulae on Binomial Convolutions of Hagen-Rothe Type." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 591-605. <http://eudml.org/doc/294019>.

@article{Chu2013,
abstract = {By means of duplicate inverse series relations, we investigate dual relations of four binomial convolution identities. Four classes of reciprocal formulae on binomial convolutions of Hagen-Rothe type are established. They reflect certain “reciprocity” on the Hagen-Rothe-like convolutions in the sense that each binomial summation involved has no closed form in general, but their sum and difference in pairs do have simple expressions in a single term of binomial coefficients.},
author = {Chu, Wenchang},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {591-605},
publisher = {Unione Matematica Italiana},
title = {Reciprocal Formulae on Binomial Convolutions of Hagen-Rothe Type},
url = {http://eudml.org/doc/294019},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Chu, Wenchang
TI - Reciprocal Formulae on Binomial Convolutions of Hagen-Rothe Type
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 591
EP - 605
AB - By means of duplicate inverse series relations, we investigate dual relations of four binomial convolution identities. Four classes of reciprocal formulae on binomial convolutions of Hagen-Rothe type are established. They reflect certain “reciprocity” on the Hagen-Rothe-like convolutions in the sense that each binomial summation involved has no closed form in general, but their sum and difference in pairs do have simple expressions in a single term of binomial coefficients.
LA - eng
UR - http://eudml.org/doc/294019
ER -

References

top
  1. ANDREWS, G. E. - BURGE, W. H., Determinant identities, Pacific J. Math., 158, 1 (1993), 1-14. Zbl0793.15001MR1200825
  2. CHU, W., An Algebraic Identity and Its Application: The unification of some matrix inverse pairs, Pure Math. & Appl., 4, 2 (1993), 175-190. Zbl0795.05015MR1270427
  3. CHU, W., Inversion Techniques and Combinatorial Identities: A quick introduction to hypergeometric evaluations, Math. Appl., 283 (1994), 31-57. Zbl0830.05006MR1292876
  4. CHU, W., Binomial convolutions and determinant identities, Discrete Math., 204, 1-3 (1999), 129-153. MR1691866DOI10.1016/S0012-365X(98)00368-9
  5. CHU, W., Some binomial convolution formulas, Fibonacci Quarterly, 40, 1 (2002), 19- 32. Zbl0999.05002MR1885266
  6. CHU, W., Duplicate Inverse Series Relations and Hypergeometric Evaluations with z = 1 / 4 , Boll. Un. Mat. Ital., B-7 (2002, Serie VIII), 585-604. Zbl1097.33003MR1934369
  7. CHU, W., Inversion Techniques and Combinatorial Identities: Balanced Hypergeo- metric Series, Rocky Mountain J. of Math., 32, 2 (2002), 561-587. Zbl1038.33002MR2088091
  8. CHU, W. - HSU, L. C., Some new applications of Gould-Hsu inversions, J. Combinatorics, Information & System Sciences, 14, 1 (1990), 1-4. Zbl0717.11007MR1068649
  9. CHU, W. - WEI, C. A., Legendre Inversions and Balanced Hypergeometric Series Identities, Discrete Mathematics, 308, 4 (2008), 541-549. Zbl1246.05016MR2376476DOI10.1016/j.disc.2007.03.031
  10. GOULD, H. W., Some generalizations of Vandermonde's convolution, Amer. Math. Month., 63, 1 (1956), 84-91. Zbl0072.00702MR75170DOI10.2307/2306429
  11. GOULD, H. W. - HSU, L. C., Some new inverse series relationsDuke Math. J., 40 (1973), 885-891. Zbl0281.05008MR337652
  12. GRAHAM, R. L. - KNUTH, D. E. - PATASHNIK, O., Concrete Mathematics, Addison-Wesley Publ. Company, Reading, Massachusetts, 1989. MR1001562
  13. MERLINI, D. - SPRUGNOLI, R. - VERRI, M. C., Combinatorial sums and implicit Riordan arrays, Discrete Mathematics, 309 (2009), 475-486. Zbl1157.05005MR2473094DOI10.1016/j.disc.2007.12.039
  14. RIORDAN, J., Combinatorial Identities, John Wiley & Sons, Inc.1968. MR231725
  15. SPRUGNOLI, R., Riordan arrays and the Abel-Gould identity, Discrete Math., 142 (1995), 213-233. MR1341448DOI10.1016/0012-365X(93)E0220-X

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.