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Hecke–Rogers, Andrews identities; combinatorial proofs

J.T. Joichi
1990 Discrete Mathematics  
An involution on the set of pairs of partitions of integers into distinct parts is given which proves an identity of Hecke-Rogers. A bijection shows equivalence with an identity of Andrews.  ...  Andrews [3] gave the identity (,G (1 -q'))2 = nzO (-l)"q'ql -q%+2) i: qk(n--k! In this paper we first give a bijection which proves the equality of the right hand sides of (3) and (4).  ...  Introduction Euler's Pentagonal Number Theorem states that fi (1 _ 4j) = 1 + $I (_l)k(p3k-lW + p3k+lP)_ j=l (1) Franklin [4] gave a combinatorial proof of this theorem.  ... 
doi:10.1016/0012-365x(90)90131-z fatcat:ijjtwbul3nf6hnpgz6kpk6t2lq

Page 410 of Mathematical Reviews Vol. , Issue 81B [page]

1981 Mathematical Reviews  
Franklin’s combinatorial proof of Euler’s pentagonal theorem, and states the author’s elegant generalization of the Rogers-Ramanu- jan identities involving successive Durfee squares [the author, Amer.  ...  Math. 77 (1978), no. 1, 71-74; MR 58 #21925] giving combinatorial proofs and generalizations of identi- ties of Rogers, and of B. Richmond and G.  ... 

Page 5312 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
Andrews, On Ramanujan’s empirical calculation for the Rogers-Ramanujan identities (27-35); Paul T. Bateman, Integers expressible in a given number of ways as a sum of two squares (37-45); Bruce C.  ...  Newman], A “natural” proof of the nonvanishing of L-series (495-498); Andrew M. Odlyzko and Chris M.  ... 

Page 5802 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
{For the entire collection see MR 91h:00024.} 91k:05012 OSAI7 11P83 Joichi, J.T. (1-MN) Hecke-Rogers, Andrews identities; combinatorial proofs. Discrete Math. 84 (1990), no. 3, 255-259.  ...  The author gives combinatorial proofs of the double series expan- sions of [](1—q’)*, 1 < j <0, found by Andrews, Hecke and Rogers. The technique is an adaptation of that used by the au- thor and D.  ... 

Author index volume 84 (1990)

1990 Discrete Mathematics  
., Hecke-Rogers, Andrews identities; combinatorial proofs (3) 255-259 Kessler, O., see Aharoni, R.  ... 
doi:10.1016/0012-365x(90)90141-4 fatcat:hp5bwqzx3bapnmbeavvedpysma

Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions [article]

Dandan Chen, Rong Chen
2021 arXiv   pre-print
We find the Hecke-Rogers type series representations of generating functions of the Hurwitz class numbers which is very close to certain mock theta functions.  ...  We also prove two combinatorial interpretation of Hurwitz class numbers appeared on OEIS.  ...  Now we can prove the Hecke-Rogers identity. Theorem 3.14. (3.24) J 2 1 J 2 F 12,−1 (q) = 1−|n|≤j≤|n| sg(n)(−1) j−1 (4n − 1)q 4n 2 −2n−3j 2 +2j . Proof.  ... 
arXiv:2107.04809v1 fatcat:36aq6x4u3jfubo3haft4ps6cpi

Mock modular forms and d -distinct partitions

Amanda Folsom
2014 Advances in Mathematics  
ordinary modular form, and the celebrated Rogers-Ramanujan identities imply that the 2-distinct parts function q −1/60 R 2 (1; q) is an ordinary modular form.  ...  Within combinatorial number theory, this function has long since been of historical importance: a famous identity of Euler and Sylvester implies that the distinct parts function q 1/24 R 1 (1; q) is an  ...  Similar to (1.1), Rogers and Ramanujan proved that R 2 (1; q) := n 0 q n 2 (q; q) n = 1 (q; q 5 ) ∞ (q 4 ; q 5 ) ∞ . (1.2) The second equality in (1.2) is one of the famous Rogers-Ramanujan identities  ... 
doi:10.1016/j.aim.2013.12.018 fatcat:lstdb76bzvfgxgmyqpxv5du6si

Forthcoming papers

1990 Discrete Mathematics  
Joichi, Hecke-Rogers, Andrews identities: combinatorial proofs. An involution on the set of pairs of partitions of integers into distinct parts is given which proves an identity of Hecke-Rogers.  ...  A bijection shows equivalence with an identity of Andrews. A.V. Karzanov, Packings of cuts realizing distances between certain vertices in a planar graph. Recently, A.  ... 
doi:10.1016/0012-365x(90)90302-x fatcat:ykrz45e3k5agpheqr3fiwuatx4

Hall-Littlewood Functions, Plane Partitions, and the Rogers-Ramanujan Identities

John R. Stembridge
1990 Transactions of the American Mathematical Society  
We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due  ...  Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series.  ...  There have also been many combinatorial generalizations of the Rogers-Ramanujan identities, beginning with Gordon [Gl] , and later with Andrews [Al, A3] and .  ... 
doi:10.2307/2001250 fatcat:m632t2vmpzamfab2jrawojpnrq

Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities

John R. Stembridge
1990 Transactions of the American Mathematical Society  
We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due  ...  Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series.  ...  There have also been many combinatorial generalizations of the Rogers-Ramanujan identities, beginning with Gordon [Gl] , and later with Andrews [Al, A3] and .  ... 
doi:10.1090/s0002-9947-1990-0986702-5 fatcat:btxnuvelrze5femevpt6ncz3vu

Applications of Generalized q-Difference Equations for General q-Polynomials

Zeya Jia, Bilal Khan, Qiuxia Hu, Dawei Niu
2021 Symmetry  
Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(∞,−1,1,q) is the generation of the fifth-order mock theta functions  ...  Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities.  ...  Furthermore, Andrews [2] also gave a new natural interpretation of the fifth-order mock theta functions along with a new proof of the Hecke-type series representation.  ... 
doi:10.3390/sym13071222 fatcat:rcbamonuorgyjfcslwc7m7ukre

Some formulae for coefficients in restricted q-products [article]

Ankush Goswami, Venkata Raghu Tej Pantangi
2020 arXiv   pre-print
By specializing N to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients.  ...  These representations were first obtained by Rogers [11] but systematically studied much later by Hecke [15] .  ...  However in some cases, the q-product in (1.1) can be expressed as a double series representation of the Hecke-Rogers type (see [2] ).  ... 
arXiv:2005.01067v2 fatcat:d6zdqxxsxjavnak55vfz6zulle

Some formulae for coefficients in restricted q-products

Ankush Goswami, Venkata Raghu Tej Pantangi
2020 Journal of Number Theory  
By specializing N to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients.  ...  Theorem 1.3 (Hecke, Rogers). We have ∞ j=1 (1 − q j ) 2 = ∞ n=0 −n/2≤m≤n/2 (−1) n+m q (n 2 −3m 2 )/2+(n+m)/2 .  ...  These representations were first obtained by Rogers [15] but systematically studied much later by Hecke [8] .  ... 
doi:10.1016/j.jnt.2020.09.008 fatcat:a2ou7fao55bybhc43t45yri32a

The A_2n^(2) Rogers-Ramanujan identities [article]

S. Ole Warnaar
2013 arXiv   pre-print
The famous Rogers-Ramanujan and Andrews--Gordon identities are embedded in a doubly-infinite family of Rogers-Ramanujan-type identities labelled by positive integers m and n.  ...  Rogers-Ramanujan-type identities for even moduli, corresponding to the affine Lie algebras C_n^(1) and D_n+1^(2), are also proven.  ...  The famous Rogers-Ramanujan and Andrews-Gordon identities are embedded in a doubly-infinite family of Rogers-Ramanujantype identities labelled by positive integers m and n.  ... 
arXiv:1309.5216v2 fatcat:lkc3vabdfneyzh4vqny2mb5ffe

Tenth order mock theta functions in Ramanujan's Lost Notebook

Youn-Seo Choi
1999 Inventiones Mathematicae  
Note that $G(q)$ and $H(q)$ which occur in the Rogers-ffimanujan identities [7]. In [5] , Andrews and F.  ...  These are based on combinatorial interpretations of Hecke type series for sixth and tenth order mock theta functions.  ... 
doi:10.1007/s002220050318 fatcat:x5sscnozc5ghpc3ijh3beqtry4
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