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q-Extensions of identities of Abel-Rothe type

1996
*
Discrete Mathematics
*

Our two main results are a commutative

doi:10.1016/0012-365x(95)00108-9
fatcat:ykfxs3ajcfhqvidbfqvzaj2eo4
*q*-analogue*of*Rothe's*identity*with an extra parameter, and a noncommutative symmetric*q*-*Abel**identity*with two extra parameters. ... A particularly nice one, essentially due to*Rothe*, is that the polynomials a,(x; h,w)= ao(x; h, w) = l, are*of*binomial*type*. ... Acknowledgements It would be difficult to overstate the impact*of*Professor G. Andrews' remark, quoted in the text, on this paper. It brought a modicum*of*structure to the mass*of*Rothe's work. ...##
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Abel-Rothe type generalizations of Jacobi's triple product identity
[article]

2003
*
arXiv
*
pre-print

Further, we apply the same method to our previous

arXiv:math/0302270v4
fatcat:3fcwzeaedvapjpkfrd3sboiq3i
*q*-*Abel*-*Rothe*summation to obtain, for the first time,*Abel*-*Rothe**type*generalizations*of*Jacobi's triple product*identity*. ... Using a simple classical method we derive bilateral series*identities*from terminating ones. In particular, we show how to deduce Ramanujan's 1-psi-1 summation from the*q*-Pfaff-Saalschuetz summation. ... In particular, one*of*the referees asked me to find*Abel*-*Rothe**type**extensions**of*the Macdonald*identities*. I thus included such an*extension*, see (5.6) . ...##
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A new multidimensional matrix inverse with applications to multiple q-series

1999
*
Discrete Mathematics
*

As applications

doi:10.1016/s0012-365x(98)00374-4
fatcat:sp23hhzaobguxf4wif4ic6eqva
*of*our matrix inversion we derive some multidimensional*q*-series*identities*. Among these are*q*-analogues*of*Carlitz' multidimensional*Abel*-*type*expansion formulas. ... Finite forms*of**identities**of*Rogers-Ramanujan*type*were considered by Bressoud [6] . ... Here is our other*extension**of*(7.4): Theorem Multiple*q*-*Abel*and*q*-*Rothe*summations We can use the expansions (7.3) and (7.4) to obtain terminating*q*-*Abel*and*q*-*Rothe*summations, respectively (for ...##
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Finite Differences and Terminating Hypergeometric Series

2016
*
Irish Mathematical Society Bulletin
*

Strehl,

doi:10.33232/bims.0078.31.45
fatcat:ukbsfhbp3nfrpbhu3uv2n72o6i
*Identities**of**Rothe*–*Abel*–Schläfli–Hurwitz–*type*, Discrete Mathe- matics 99:1–3 (1992), 321–340. [25] A. Verma, V. K. Jain and S. ... Chu, Elementary Proofs for Convolution*Identities**of**Abel*and Hagen–*Rothe*, Electronic Journal*of*Combinatorics 17 (2010), #N24. [11] L. ...##
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Two matrix inversions associated with the Hagen–Rothe formula, their q-analogues and applications

2011
*
Journal of combinatorial theory. Series A
*

Some related

doi:10.1016/j.jcta.2010.12.012
fatcat:baoxygvti5glzgcl73cnxhgtrm
*q*-series inverse relations using the known*q*-analogues*of*the Hagen-*Rothe*formula are established. ... These new inversions uniformize Riordan's inverse relations*of**Abel*-, Chebyshev-, and Legendre-*type*as well as Gould's inversions based on Vandermonde-*type*convolutions. ... Their informative indications to*q*-analogues*of*the Hagen-*Rothe*formula ultimately led the author to the discovery*of*Theorem 3. Thanks also go to Professor W. Chu for his constant support. ...##
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Page 5972 of Mathematical Reviews Vol. , Issue 97J
[page]

1997
*
Mathematical Reviews
*

Christian Krattenthaler (A-VUC; Vienna)
97j:05015 05A30 11B65 Johnson, Warren P. (1-PAS; University Park, PA)

*q*-*extensions**of**identities**of**Abel*-*Rothe**type*. ... The author finds a very general family*of*polynomials*of**q*- binomial*type*, n—| a,(x;b,h,w,*q*) =(x +b) [ [G4 +b+[ijh+[njw), i=l where [i] = (1 — g‘)/(1 —*q*). ...##
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Page 5446 of Mathematical Reviews Vol. , Issue 2001H
[page]

2001
*
Mathematical Reviews
*

As applications multidimensional

*identities**of**q*-*Abel*and*q*-*Rothe**type*are derived. {For the entire collection see MR 2001a:33001.} Peter J. ... In the one-dimensional case the*q*-hypergeometric integrals can be expressed in terms*of*the basic*q*-hypergeometric series and the corresponding*identity*re- duces to bilinear*identities*for this series ...##
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Multinomial convolution polynomials

1996
*
Discrete Mathematics
*

In [9] Knuth shows how to derive the convolution formulas

doi:10.1016/0012-365x(95)00160-x
fatcat:xairizcw5bgmjpgkbaxmzp4c4a
*of*Hagen,*Rothe*and*Abel*from Vandermonde's convolution and the binomial theorem for integer exponents. ... In the present paper, we shall first present a short and elementary proof*of*the multi-*extension**of*the above convolution formulas, due to Raney and Mohanty. ... -k(y). k-OThe two typical families*of*convolution polynomials are the binomial theorem*extensions**of*the above*identities*have been given by*Abel*,*Rothe*and Hagen in the last century. ...##
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Page 1215 of Mathematical Reviews Vol. , Issue 93c
[page]

1993
*
Mathematical Reviews
*

The cardinality

*of*these two sets is just the nth Catalan number.” 93c:05010 O5A15 05A19 Strehl, Volker (D-ERL-I1)*Identities**of**Rothe*-*Abel*-Schlafli-Hurwitz-*type*. ... The author provides a simple combinatorial construction from which many well-known convolution*identities*, such as those by*Abel*,*Rothe*, Schlafli and Hurwitz, and their*extensions*and gener- alizations ...##
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Page 52 of Mathematical Reviews Vol. , Issue 90A
[page]

1990
*
Mathematical Reviews
*

to Hagen-

*Rothe**identities*, respectively. ... Wan Di Wei (PRC-SICH) 90a:05018 05A19 05430 Denis, Remy Y. (6-GORA) On certain summation*of**q*-series and*identities**of*Rogers-Ramanujan*type*. J. Math. Phys. Sci. 22 (1988), no. 1, 87-99. ...##
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Page 1246 of Mathematical Reviews Vol. , Issue 92c
[page]

1992
*
Mathematical Reviews
*

As examples, the

*Abel**identity*, the Hagen-*Rothe*formula and the corresponding series transformations are derived immediately.” 92c:05019 05A20 Bollobas, B. (4-CAMB); Radcliffe, A. ... Summary: “A mechanical technique is developed for the proof*of*combinatorial*identities*by means*of*Gould-Hsu inversion. ...##
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Curious extensions of Ramanujan's ψ11 summation formula

2007
*
Journal of Mathematical Analysis and Applications
*

We deduce new

doi:10.1016/j.jmaa.2006.12.074
fatcat:ba6rmavfqvhy5nd7bnlxy4zfcq
*q*-series*identities*by applying inverse relations to certain*identities*for basic hypergeometric series. ... The*identities*obtained themselves do not belong to the hierarchy*of*basic hypergeometric series. ... In [9] also multidimensional*identities*associated with root systems*of**Abel*-,*Rothe*-and the above "curious"-*type*are derived. Related beta*type*integrals are deduced in [4] and [10] . ...##
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New Curious Bilateral q-Series Identities

2012
*
Axioms
*

By applying a classical method, already employed by Cauchy, to a terminating curious summation by one

doi:10.3390/axioms1030365
fatcat:yfunjlhs7ffklnsphdi3bfhpam
*of*the authors, a new curious bilateral*q*-series*identity*is derived. ... Acknowledgments The second author was partly supported by FWF Austrian Science Fund grant S9607 (which is part*of*the Austrian national Research Network "Analytic Combinatorics and probabilistic Number ... Another method to obtain bilateral summations from terminating ones was employed in [4] to give a new proof*of*Ramanujan's 1 ψ 1 summation formula and to derive (for the first time)*Abel*-*Rothe**type**extensions*...##
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On some classes of inverse series relations and their applications

1993
*
Discrete Mathematics
*

This paper gives a brief exposition

doi:10.1016/0012-365x(93)90003-c
fatcat:uvecdiou3jdu5ifx6u5t4jhi4q
*of*several recent results obtained by the authors concerning some classes*of*inverse relations, including the general binomial-*type*inversions and two kinds*of*general ... Also described are some applications*of*the related inversion techniques to combinatorics (including new proofs*of*Rogers-Ramanujan*identities*and MacMahon's master theorem), interpolation methods and ... In particular, it has been shown that the classical*identities*due to*Abel*, Hagen-*Rothe*, Jensen, Rohatgi, Moriarty, Van Ebbenhorst-Tengbergen all belong to CuC*, so that they can be proved rather straightforwardly ...##
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Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model)
[article]

2009
*
arXiv
*
pre-print

form by Leroux and Gessel, and an

arXiv:0803.1477v2
fatcat:xpqcjaucafellaxvqqjwiwm56y
*identity*for the inversion enumerator*of*trees found by Mallows, Riordan and Kreweras. ... We also prove some further*identities*for the multivariate Tutte polynomial, which generalize an*identity*for counting connected graphs found by Riordan, Nijenhuis, Wilf and Kreweras and in more general ... Finally, we are grateful to an anonymous referee for pointing out the connection*of*the formulae*of*Section 3.3.1 with Möbius inversion -which led us to discover the results reported in the Appendix -and ...
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