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Hecke–Rogers, Andrews identities; combinatorial proofs

1990
*
Discrete Mathematics
*

An involution on the set of pairs of partitions of integers into distinct parts is given which proves an

doi:10.1016/0012-365x(90)90131-z
fatcat:ijjtwbul3nf6hnpgz6kpk6t2lq
*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. ...##
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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. ...##
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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. ...

##
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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. ...##
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Author index volume 84 (1990)

1990
*
Discrete Mathematics
*

.,

doi:10.1016/0012-365x(90)90141-4
fatcat:hp5bwqzx3bapnmbeavvedpysma
*Hecke*-*Rogers*,*Andrews**identities*;*combinatorial**proofs*(3) 255-259 Kessler, O., see Aharoni, R. ...##
###
Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions
[article]

2021
*
arXiv
*
pre-print

We find the

arXiv:2107.04809v1
fatcat:36aq6x4u3jfubo3haft4ps6cpi
*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*. ...##
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Mock modular forms and d -distinct partitions

2014
*
Advances in Mathematics
*

ordinary modular form, and the celebrated

doi:10.1016/j.aim.2013.12.018
fatcat:lstdb76bzvfgxgmyqpxv5du6si
*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*...##
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Forthcoming papers

1990
*
Discrete Mathematics
*

Joichi,

doi:10.1016/0012-365x(90)90302-x
fatcat:ykrz45e3k5agpheqr3fiwuatx4
*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. ...##
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Hall-Littlewood Functions, Plane Partitions, and the Rogers-Ramanujan Identities

1990
*
Transactions of the American Mathematical Society
*

We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series

doi:10.2307/2001250
fatcat:m632t2vmpzamfab2jrawojpnrq
*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 . ...##
###
Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities

1990
*
Transactions of the American Mathematical Society
*

We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series

doi:10.1090/s0002-9947-1990-0986702-5
fatcat:btxnuvelrze5femevpt6ncz3vu
*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 . ...##
###
Applications of Generalized q-Difference Equations for General q-Polynomials

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. ...

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

2020
*
arXiv
*
pre-print

By specializing N to different values, we see that these expressions simplify in some cases and we obtain several nice

arXiv:2005.01067v2
fatcat:d6zdqxxsxjavnak55vfz6zulle
*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] ). ...##
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Some formulae for coefficients in restricted q-products

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

doi:10.1016/j.jnt.2020.09.008
fatcat:a2ou7fao55bybhc43t45yri32a
*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] . ...##
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The A_2n^(2) Rogers-Ramanujan identities
[article]

2013
*
arXiv
*
pre-print

The famous

arXiv:1309.5216v2
fatcat:lkc3vabdfneyzh4vqny2mb5ffe
*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. ...##
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Tenth order mock theta functions in Ramanujan's Lost Notebook

1999
*
Inventiones Mathematicae
*

Note that $G(q)$ and $H(q)$ which occur in the

doi:10.1007/s002220050318
fatcat:x5sscnozc5ghpc3ijh3beqtry4
*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. ...
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