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q-Extensions of identities of Abel-Rothe type

Warren P. Johnson
1996 Discrete Mathematics  
Our two main results are a commutative 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.  ... 
doi:10.1016/0012-365x(95)00108-9 fatcat:ykfxs3ajcfhqvidbfqvzaj2eo4

Abel-Rothe type generalizations of Jacobi's triple product identity [article]

Michael J. Schlosser
2003 arXiv   pre-print
Further, we apply the same method to our previous 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) .  ... 
arXiv:math/0302270v4 fatcat:3fcwzeaedvapjpkfrd3sboiq3i

A new multidimensional matrix inverse with applications to multiple q-series

Christian Krattenthaler, Michael Schlosser
1999 Discrete Mathematics  
As applications 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  ... 
doi:10.1016/s0012-365x(98)00374-4 fatcat:sp23hhzaobguxf4wif4ic6eqva

Finite Differences and Terminating Hypergeometric Series

W. Chu
2016 Irish Mathematical Society Bulletin  
Strehl, Identities of RotheAbel–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.  ... 
doi:10.33232/bims.0078.31.45 fatcat:ukbsfhbp3nfrpbhu3uv2n72o6i

Two matrix inversions associated with the Hagen–Rothe formula, their q-analogues and applications

Xinrong Ma
2011 Journal of combinatorial theory. Series A  
Some related 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.  ... 
doi:10.1016/j.jcta.2010.12.012 fatcat:baoxygvti5glzgcl73cnxhgtrm

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).  ... 

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  ... 

Multinomial convolution polynomials

Jiang Zeng
1996 Discrete Mathematics  
In [9] Knuth shows how to derive the convolution formulas 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.  ... 
doi:10.1016/0012-365x(95)00160-x fatcat:xairizcw5bgmjpgkbaxmzp4c4a

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  ... 

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.  ... 

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.  ... 

Curious extensions of Ramanujan's ψ11 summation formula

Victor J.W. Guo, Michael J. Schlosser
2007 Journal of Mathematical Analysis and Applications  
We deduce new 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] .  ... 
doi:10.1016/j.jmaa.2006.12.074 fatcat:ba6rmavfqvhy5nd7bnlxy4zfcq

New Curious Bilateral q-Series Identities

Frédéric Jouhet, Michael J. Schlosser
2012 Axioms  
By applying a classical method, already employed by Cauchy, to a terminating curious summation by one 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  ... 
doi:10.3390/axioms1030365 fatcat:yfunjlhs7ffklnsphdi3bfhpam

On some classes of inverse series relations and their applications

Wenchang Chu, Leetsch C. Hsu
1993 Discrete Mathematics  
This paper gives a brief exposition 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  ... 
doi:10.1016/0012-365x(93)90003-c fatcat:uvecdiou3jdu5ifx6u5t4jhi4q

Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model) [article]

Alexander D. Scott, Alan D. Sokal
2009 arXiv   pre-print
form by Leroux and Gessel, and an 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  ... 
arXiv:0803.1477v2 fatcat:xpqcjaucafellaxvqqjwiwm56y
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