(a; q),~x"_= a Ot(.;~,~, ) (1) (a; q)== l~i (1- aq"), rt=0
|
|
- Angelica Crawford
- 5 years ago
- Views:
Transcription
1 Aequationes Mathematicae 18 (1978) University of Waterloo Birkh~user Verlag, Basel A simple proof of Ramanujan's summation of the 1~1 GEORGE E. ANDREWS and RICHARD ASKEY Abstract. A simple proof by functional equations is given for Ramanujan's 14'1 sum. Ramanujan's sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series. In [5;p. 222, eq.( )] G. H. Hardy alludes to Ramanujan's "... remarkable formula with many parameters.": where (a; q),~x"_= a Ot(.;~,~, ) (1)... (b; q). = (b/a; q) (q;q)~(q/ax; q)~(ax; q)~ (b; q)~(b/ax; q)~(q/a; q)~(x; q)~ ' Ib <,x,<l,,q,<l, (a; q)== l~i (1- aq"), rt=0 and (a; q). = (a; q)=/(aq"; ql. There are four published proofs of this result ([1], [2], [4] and [7]). Those in [1], [2] and [7] rely on somewhat tricky rearrangement of series and on the q-analog of Gauss's summation [10; p. 97, eq. ( )] X~ (a; q).(b; q).(c/ab)" (c/a; q)=(c/b; q)=.=o ~ (c; q).(q; a). (c ; q)=(c/ab ; q)= " (2) AMS (1970) subject classification: Primary 33A25, 33A30. The work of the first author was partially sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and NSF Grant and that of the second author was partially supported by NSF Grant MPS AO2. Received August 23, 1976 and, in ]inal form, February i7,
2 334 GEORGE E. ANDREWS AND RICHARD ASKEY AEO. MATH. where [c[<min(1,[ab[). The other proof uses the q-analogue of the binomial series [10; p. 92, eq. ( )]: (a;q)"t"=(at;q) Itl< 1, Iql< 1, (3),,=o (q; q),, (t; q)~ ' but it is far from simple. Since Ramanujan's summation (1) has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series [3], it has become worthwhile to present an almost trivial proof of (1). Another very simple proof has been found by M. Ismail [6]. Proof of (1). We begin by noting that for Iql< 1. f(b)= ~x(a;~ 'x) is an analytic function of b inside [b[<min(1, lax[), since f(b) = ~, (a;q),,xn+ ~, (1-b/q")'".(1-b/q)x-".=o (b; q)..~_-a il--~'" : ( a-~q) " (4) Furthermore, (a, q),,+lx ll//l(a;~ 'x)- a 1~(";~ "q~) =,=_ (b; q), =x-l(l-b) ~ (a;q)"+lx"+x Hence ~. (a, q),q"x" f(bq)- x-x(1 - b)f(b) = a, =- (bq; q), (6) =_ab_l ~ (a;q),(1-bq"-l)x"... (bq; q),, = - ab-x(1 - b)f(b) + ab-lf(bq), and so
3 Vol. 18, 1978 A simple proof of Ramanujan's summation of the ~ g'~ 335 or (1 - b]a) f(b) = (1 - b)(1 - b/ax) f(bq). (7) If we iterate (7) n- 1 times, we find that (bla ; q). f(b) = f(bq"), (8) (b; q),.(b/ax; q). and, since f(b) is analytic in the neighborhood of 0 given by Ibl < lag we obtain in the limit as n ~ 0o, (b/a; qlf(o) f(b) = (b; q)=(b/ax; q)= " (9) Now we observe from (4) and (3) that f(q) = ~. (a; q).x"= (ax, q)= (10).=o (q; q). (x; ql " This allows us to evaluate f(0) by setting b =q in (9): f(o) = (q; q)~(q/ax; q)~f(q) (q/a; q)~o (q; q)~(~x q)oo (ax; q)~ (q/a; q)~(x; q)~ Finally we may utilize (11) to eliminate f(0) from (9): (11) (b/a; q)=(q; q)=(q/ax; q)~(ax; q) 1~,( ~ "~) = f(b) = (b; q) (b/ax; q)~(q/a, q)oo(x; q)~ ' (12) as desired. Note that Jacobi's triple product identity follows directly from (1) if we replace a byoe -1,x by za and then set a=b=0: ~. (-1)"q"("-l)/2z" = (q; ql(q/z; q)~(z; q)~. (13) n=-oo
4 336 GEORGE E. ANDREWS AND RICHARD ASKEY AEO. MATH. I. J. Schoenberg has pointed out an interesting property of (a; q)d(b; q),, which follows from Ramanujan's sum. A sequence a,, n = 0, +1,..., is said to be totally positive if all subdeterminants of the doubly infinite matrix A= (a~_j)_=<i,j<= are nonnegative. Schoenberg [9] proved that a sequence an is totally positive if the bilateral generating function f(z) = ~.~_~ a,z '~ has the representation 1 l~l (1 + a~z)(1 + 8,z -1) f(z) ecz+dz,=,ix (1 -/3~z)(1 - -/,z- ~) ' (14) c, d, a~,/3~, 3'~, 1~ -> O, ~ (a~ +/3~ + y~ + 8~) < oo, i=l in the interior of an annulus centered at the origin. If a < b <0 in (1) then, the generating function has the form (14) and so _ (a; q). _ fl (1 - bqk+")(1 - aq k) a. (b;q). k=o(1--aqk+")(1--bq k) is a totally positive sequence for a < b =<0, 0< q < 1. Schoenberg [9] proved this when b = 0. For an extended discussion of totally positive sequences see Karlin [83. REFERENCES [1] ANDREWS, G. E., On Ramanujan's summation of l~l(a;b;z). Proc. Amer. Math. Soc. 22 (1969), [21 ANDREWS, G. E., On a transformation of bilateral series with applications. Proc. Amer. Math. Soc. 25 (1970), [3] ANDREWS, G. E. and ASKEY, R., The classical and discrete orthogonal polynomials and their q-analogues. To appear. [4] HAHN, W., Beitriige zur Theorie der Heineschen Reihen. Math. Nach. 2 (1949), [5] HARDY, G. H., Ramanujan. Cambridge University Press, Cambridge, Reprinted: Chelsea, New York. [6] ISMAIL, M., A simple proof of Ramanulan's l~bl sum. Proc. Amer. Math. Scc., 63 (1977), [7] JACKSON, M., On Lerch's transcendant and the basic bilateral hypergeometric series 2~z- J. London Math. Soc 25 (1950), [8] KARLIN, S., Total l~ositivity, Volume One. Stanford University Press, Stanford, 1968.
5 Vol. 18, 1978 A simple proof of Ramanujan's summation of the t~l 337 [9] SCHOENBERG, I. J., Some analytical aspects o[ the problem o[ smoothing. Studies and Essays presented to R. Courant on his 60th Birthday, Interscience Publishers, New York, 1948, [10] SLATER, L. J., Generalized hypergeometric [unctions. Cambridge University Press, Cambridge Pennsylvania State University University Park, Pennsylvania U.S.A. University of Wisconsin, Madison, Wisconsin U.S.A.
I I I I. r,. ta,s. ~~~~~~~~ osz 939 wzs!sin univ MAoIS ~~~~ MAT ~~~ AT RcScARcH CcwTzp. t.iu ~ yn ~ LASSZFflD
r osz 939 wzs!sn unv MAoS MAT AT RcScARcH CcwTzp r,. ta,s PROOF OF sa MANUJAN flumm*t ON OF NC Sue S PS Sub t ANuRES. t.iu yn LASSZFflD N llu h 2. mu 25 llhi M risi All,,1, AllU.4 1 MPLE J ROOF OF AMANUJAN
More informationCOMBINATORIAL PROOFS OF RAMANUJAN S 1 ψ 1 SUMMATION AND THE q-gauss SUMMATION
COMBINATORIAL PROOFS OF RAMANUJAN S 1 ψ 1 SUMMATION AND THE q-gauss SUMMATION AE JA YEE 1 Abstract. Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities
More informationOn the 3 ψ 3 Basic. Bilateral Hypergeometric Series Summation Formulas
International JMath Combin Vol4 (2009), 41-48 On the 3 ψ 3 Basic Bilateral Hypergeometric Series Summation Formulas K RVasuki and GSharath (Department of Studies in Mathematics, University of Mysore, Manasagangotri,
More informationHENG HUAT CHAN, SONG HENG CHAN AND SHAUN COOPER
THE q-binomial THEOREM HENG HUAT CHAN, SONG HENG CHAN AND SHAUN COOPER Abstract We prove the infinite q-binomial theorem as a consequence of the finite q-binomial theorem 1 The Finite q-binomial Theorem
More informationThe Abel Lemma and the q-gosper Algorithm
The Abel Lemma and the q-gosper Algorithm Vincent Y. B. Chen, William Y. C. Chen 2, and Nancy S. S. Gu 3 Center for Combinatorics, LPMC Nankai University, Tianjin 30007 P. R. China Email: ybchen@mail.nankai.edu.cn,
More informationBASIC HYPERGEOMETRIC SERIES
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS BASIC HYPERGEOMETRIC SERIES Second Edition GEORGE GASPER Northwestern University, Evanston, Illinois, USA MIZAN RAHMAN Carleton University, Ottawa, Canada
More informationNEW CURIOUS BILATERAL q-series IDENTITIES
NEW CURIOUS BILATERAL q-series IDENTITIES FRÉDÉRIC JOUHET AND MICHAEL J. SCHLOSSER Abstract. By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the
More informationarxiv:math/ v1 [math.nt] 28 Jan 2005
arxiv:math/0501528v1 [math.nt] 28 Jan 2005 TRANSFORMATIONS OF RAMANUJAN S SUMMATION FORMULA AND ITS APPLICATIONS Chandrashekar Adiga 1 and N.Anitha 2 Department of Studies in Mathematics University of
More informationSome More Identities of Rogers-Ramanujan Type
Georgia Southern University Digital Commons@Georgia Southern Mathematical Sciences Faculty Publications Department of Mathematical Sciences 2009 Some More Identities of Rogers-Ramanujan Type Douglas Bowman
More informationCombinatorial Analysis of the Geometric Series
Combinatorial Analysis of the Geometric Series David P. Little April 7, 205 www.math.psu.edu/dlittle Analytic Convergence of a Series The series converges analytically if and only if the sequence of partial
More informationRamanujan-Slater Type Identities Related to the Moduli 18 and 24
Ramanujan-Slater Type Identities Related to the Moduli 18 and 24 James McLaughlin Department of Mathematics, West Chester University, West Chester, PA; telephone 610-738-0585; fax 610-738-0578 Andrew V.
More informationCOMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES
COMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES James McLaughlin Department of Mathematics, West Chester University, West Chester, PA 9383, USA jmclaughl@wcupa.edu Andrew V. Sills Department of Mathematical
More informationOn integral representations of q-gamma and q beta functions
On integral representations of -gamma and beta functions arxiv:math/3232v [math.qa] 4 Feb 23 Alberto De Sole, Victor G. Kac Department of Mathematics, MIT 77 Massachusetts Avenue, Cambridge, MA 239, USA
More informationOn an identity of Gessel and Stanton and the new little Göllnitz identities
On an identity of Gessel and Stanton and the new little Göllnitz identities Carla D. Savage Dept. of Computer Science N. C. State University, Box 8206 Raleigh, NC 27695, USA savage@csc.ncsu.edu Andrew
More informationMOCK THETA FUNCTIONS AND THETA FUNCTIONS. Bhaskar Srivastava
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 36 (2007), 287 294 MOCK THETA FUNCTIONS AND THETA FUNCTIONS Bhaskar Srivastava (Received August 2004). Introduction In his last letter to Hardy, Ramanujan gave
More informationThe Bhargava-Adiga Summation and Partitions
The Bhargava-Adiga Summation and Partitions By George E. Andrews September 12, 2016 Abstract The Bhargava-Adiga summation rivals the 1 ψ 1 summation of Ramanujan in elegance. This paper is devoted to two
More information4-Shadows in q-series and the Kimberling Index
4-Shadows in q-series and the Kimberling Index By George E. Andrews May 5, 206 Abstract An elementary method in q-series, the method of 4-shadows, is introduced and applied to several poblems in q-series
More informationCONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition
More informationTHE ABEL LEMMA AND THE q-gosper ALGORITHM
MATHEMATICS OF COMPUTATION Volume 77 Number 262 April 2008 Pages 057 074 S 0025-57807)0968-0 Article electronically published on October 24 2007 THE ABEL LEMMA AND THE q-gosper ALGORITHM VINCENT Y B CHEN
More informationCOMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES. James McLaughlin Department of Mathematics, West Chester University, West Chester, PA 19383, USA
COMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES James McLaughlin Department of Mathematics, West Chester University, West Chester, PA 9383, USA jmclaughl@wcupa.edu Andrew V. Sills Department of Mathematical
More informationANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE S IDENTITY AND EULER S PENTAGONAL NUMBER THEOREM
q-hypergeometric PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE S IDENTITY AND EULER S PENTAGONAL NUMBER THEOREM S OLE WARNAAR Abstract We present alternative, q-hypergeometric
More informationThe Truncated Pentagonal Number Theorem
The Truncated Pentagonal Number Theorem George E. Andrews Department of Mathematics The Pennsylvania State University University Park, PA 16802 USA Mircea Merca Doctoral School in Applied Mathematics University
More informationTHE BAILEY TRANSFORM AND FALSE THETA FUNCTIONS
THE BAILEY TRANSFORM AND FALSE THETA FUNCTIONS GEORGE E ANDREWS 1 AND S OLE WARNAAR 2 Abstract An empirical exploration of five of Ramanujan s intriguing false theta function identities leads to unexpected
More informationA generalisation of the quintuple product identity. Abstract
A generalisation of the quintuple product identity Abstract The quintuple identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general
More information(q n a; q) = ( a) n q (n+1 2 ) (q/a; q)n (a; q). For convenience, we employ the following notation for multiple q-shifted factorial:
ARCHIVUM MATHEMATICUM (BRNO) Tomus 45 (2009) 47 58 SEVERAL q-series IDENTITIES FROM THE EULER EXPANSIONS OF (a; q) AND (a;q) Zhizheng Zhang 2 and Jizhen Yang Abstract In this paper we first give several
More informationA Fine Dream. George E. Andrews (1) January 16, 2006
A Fine Dream George E. Andrews () January 6, 2006 Abstract We shall develop further N. J. Fine s theory of three parameter non-homogeneous first order q-difference equations. The obect of our work is to
More informationCONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit, and Yee introduced partition
More information616 W. A. AL-SALAM [June
616 W. A. AL-SALAM [June 2. Colin Clark C. A. Swanson, Comparison theorems for elliptic differential equations, Proc. Amer. Math. Soc. 16 (1965), 886-890. 3. F. R. Gantmacher, The theory of matrices, Vol.
More informationDETERMINANT IDENTITIES FOR THETA FUNCTIONS
DETERMINANT IDENTITIES FOR THETA FUNCTIONS SHAUN COOPER AND PEE CHOON TOH Abstract. Two proofs of a theta function identity of R. W. Gosper and R. Schroeppel are given. A cubic analogue is presented, and
More informationTHREE-SQUARE THEOREM AS AN APPLICATION OF ANDREWS 1 IDENTITY
THREE-SQUARE THEOREM AS AN APPLICATION OF ANDREWS 1 IDENTITY S. Bhargava Department of Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, India Chandrashekar Adiga Department of Mathematics,
More informationq GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS
q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS BRUCE C. BERNDT 1 and AE JA YEE 1. Introduction Recall that the q-gauss summation theorem is given by (a; q) n (b; q) ( n c ) n (c/a; q) (c/b; q) =, (1.1)
More informationNEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE
NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE To George Andrews, who has been a great inspiration, on the occasion of his 70th birthday Abstract.
More informationJournal of Combinatorial Theory, Series A
Journal of Combinatorial Theory, Series A 118 (2011) 240 247 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series A wwwelseviercom/locate/jcta Bailey s well-poised 6 ψ 6 -series
More informationAbstract. The present paper concerns with the continued fraction representation for
italian journal of pure and applied mathematics n. 27 2010 (191 200 191 A NOTE ON CONTINUED FRACTIONS AND 3 ψ 3 SERIES Maheshwar Pathak Pankaj Srivastava Department of Mathematics Motilal Nehru National
More informationOWN ZEROS LUIS DANIEL ABREU
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 4 3 FUNCTIONS q-orthogonal WITH RESPECT TO THEIR OWN ZEROS LUIS DANIEL ABREU Abstract: In [4], G. H. Hardy proved that,
More informationTwo finite forms of Watson s quintuple product identity and matrix inversion
Two finite forms of Watson s uintuple product identity and matrix inversion X. Ma Department of Mathematics SuZhou University, SuZhou 215006, P.R.China Submitted: Jan 24, 2006; Accepted: May 27, 2006;
More informationOn Some Transformations of A 2 Formula And Their Applications
On Some Transformations of A 2 Formula And Their Applications Journal of Applied Mathematics and Computation (JAMC), 2018, 2(10), 456-465 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645
More informationInteger Partitions With Even Parts Below Odd Parts and the Mock Theta Functions
Integer Partitions With Even Parts Below Odd Parts and the Mock Theta Functions by George E. Andrews Key Words: Partitions, mock theta functions, crank AMS Classification Numbers: P84, P83, P8, 33D5 Abstract
More informationSOME IDENTITIES RELATING MOCK THETA FUNCTIONS WHICH ARE DERIVED FROM DENOMINATOR IDENTITY
Math J Okayama Univ 51 (2009, 121 131 SOME IDENTITIES RELATING MOCK THETA FUNCTIONS WHICH ARE DERIVED FROM DENOMINATOR IDENTITY Yukari SANADA Abstract We show that there exists a new connection between
More information(1 aq k ); jqj < 1; b 1 ; : : : ; b s. c0000 American Mathematical Society /00 $ $.25 per page
Fields Institute Communications Volume 00, 0000 (Nov. 20, 1995) Elementary derivations of summation and transformation formulas for q-series George Gasper Department of Mathematics Northwestern University
More informationOVERPARTITIONS AND GENERATING FUNCTIONS FOR GENERALIZED FROBENIUS PARTITIONS
OVERPARTITIONS AND GENERATING FUNCTIONS FOR GENERALIZED FROBENIUS PARTITIONS SYLVIE CORTEEL JEREMY LOVEJOY AND AE JA YEE Abstract. Generalized Frobenius partitions or F -partitions have recently played
More informationOperators with numerical range in a closed halfplane
Operators with numerical range in a closed halfplane Wai-Shun Cheung 1 Department of Mathematics, University of Hong Kong, Hong Kong, P. R. China. wshun@graduate.hku.hk Chi-Kwong Li 2 Department of Mathematics,
More informationMock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series
Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series James Mc Laughlin Abstract The bilateral series corresponding to many of the third- fifth- sixth- and eighth order moc
More informationON q-integral TRANSFORMS AND THEIR APPLICATIONS
Bulletin of Mathematical Analysis and Applications ISSN: 82-29, URL: http://www.bmathaa.org Volume 4 Issue 2 22), Pages 3-5 ON q-integral TRANSFORMS AND THEIR APPLICATIONS COMMUNICATED BY R.K. RAINA) DURMUŞ
More informationResearch Article Continued Fractions of Order Six and New Eisenstein Series Identities
Numbers, Article ID 64324, 6 pages http://dxdoiorg/055/204/64324 Research Article Continued Fractions of Order Six and New Eisenstein Series Identities Chandrashekar Adiga, A Vanitha, and M S Surekha Department
More informationA Determinant Identity that Implies Rogers-Ramanujan
A Determinant Identity that Implies Rogers-Ramanujan Kristina C. Garrett Department of Mathematics and Computer Science Carleton College, Minnesota, USA kgarrett@carleton.edu Submitted: Oct 2, 2004; Accepted:
More informationENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS. Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS Preface page xiii 1 The Gamma and Beta Functions 1 1.1 The Gamma
More informationCONSTRUCTION OF STEINER QUADRUPLE SYSTEMS HAVING LARGE NUMBERS OF NONISOMORPHIC ASSOCIATED STEINER TRIPLE SYSTEMS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, May 1975 CONSTRUCTION OF STEINER QUADRUPLE SYSTEMS HAVING LARGE NUMBERS OF NONISOMORPHIC ASSOCIATED STEINER TRIPLE SYSTEMS CHARLES
More information= i 0. a i q i. (1 aq i ).
SIEVED PARTITIO FUCTIOS AD Q-BIOMIAL COEFFICIETS Fran Garvan* and Dennis Stanton** Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved
More informationRAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS
RAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS BRUCE C. BERNDT, BYUNGCHAN KIM, AND AE JA YEE Abstract. Combinatorial proofs
More informationPositivity of Turán determinants for orthogonal polynomials
Positivity of Turán determinants for orthogonal polynomials Ryszard Szwarc Abstract The orthogonal polynomials p n satisfy Turán s inequality if p 2 n (x) p n 1 (x)p n+1 (x) 0 for n 1 and for all x in
More informationFOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS
FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS GEORGE E. ANDREWS, BRUCE C. BERNDT, SONG HENG CHAN, SUN KIM, AND AMITA MALIK. INTRODUCTION On pages and 7 in his Lost Notebook [3], Ramanujan recorded
More informationSome Restricted Plane partitions and Associated Lattice Paths S. Bedi Department of Mathematics, D.A.V College, Sector 10 Chandigarh , India
Some Restricted Plane partitions and Associated Lattice Paths S. Bedi Department of Mathematics, D.A.V College, Sector 10 Chandigarh - 160010, India Abstract. Anand and Agarwal, (Proc. Indian Acad. Sci.
More informationA q-analogue of Kummer s 20 relations
J. Math. Anal. Appl. 272 (2002 43 54 www.academicpress.com A q-analogue of Kummer s 20 relations Shaun Cooper Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private
More informationArithmetic Properties of Partition k-tuples with Odd Parts Distinct
3 7 6 3 Journal of Integer Sequences, Vol. 9 06, Article 6.5.7 Arithmetic Properties of Partition k-tuples with Odd Parts Distinct M. S. Mahadeva Naika and D. S. Gireesh Department of Mathematics Bangalore
More informationDYSON'S CRANK OF A PARTITION
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Number, April 988 DYSON'S CRANK OF A PARTITION GEORGE E. ANDREWS AND F. G. GARVAN. Introduction. In [], F. J. Dyson defined the rank
More informationNonnegative linearization for little q-laguerre polynomials and Faber basis
Journal of Computational and Applied Mathematics 199 (2007) 89 94 www.elsevier.com/locate/cam Nonnegative linearization for little q-laguerre polynomials and Faber basis Josef Obermaier a,, Ryszard Szwarc
More informationSome theorems on the explicit evaluations of singular moduli 1
General Mathematics Vol 17 No 1 009) 71 87 Some theorems on the explicit evaluations of singular moduli 1 K Sushan Bairy Abstract At scattered places in his notebooks Ramanujan recorded some theorems for
More informationA REFINEMENT OF THE ALLADI-SCHUR THEOREM
A REFINEMENT OF THE ALLADI-SCHUR THEOREM GEORGE E. ANDREWS Abstract. K. Alladi first observed a variant of I. Schur s 1926 partition theore. Namely, the number of partitions of n in which all parts are
More informationRAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS
RAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS BRUCE C. BERNDT, BYUNGCHAN KIM, AND AE JA YEE 2 Abstract. Combinatorial proofs
More informationAbstract In this paper I give an elementary proof of congruence identity p(11n + 6) 0(mod11).
An Elementary Proof that p(11n + 6) 0(mod 11) By: Syrous Marivani, Ph.D. LSU Alexandria Mathematics Department 8100 HWY 71S Alexandria, LA 71302 Email: smarivani@lsua.edu Abstract In this paper I give
More information1, if k = 0. . (1 _ qn) (1 _ qn-1) (1 _ qn+1-k) ... _: =----_...:... q--->1 (1- q) (1 - q 2 ) (1- qk) - -- n! k!(n- k)! n """"' n. k.
Abstract. We prove the ifiite q-biomial theorem as a cosequece of the fiite q-biomial theorem. 1. THE FINITE q-binomial THEOREM Let x ad q be complex umbers, (they ca be thought of as real umbers if the
More informationOn the Irrationality of Z (1/(qn+r))
JOURNAL OF NUMBER THEORY 37, 253-259 (1991) On the Irrationality of Z (1/(qn+r)) PETER B. BORWEIN* Department of Mathematics, Statistics, and Computing Science, Dalhousie University, Halifax, Nova Scotia,
More informationNEW GENERATING FUNCTIONS FOR CLASSICAL POLYNOMIALS1. J. w. brown
NEW GENERATING FUNCTIONS FOR CLASSICAL POLYNOMIALS1 J. w. brown 1. Introduction. As recently as 1951 Brafman [l]-[4] initiated a series of papers in which he gave a variety of new unusual generating functions
More informationHomepage: WWW: george/
LIST OF PUBLICATIONS of George Gasper (George Gasper, Jr.) (2/16/07 version) Department of Mathematics, Northwestern University, Evanston, Illinois 60208, (847) 491-5592 E-mail: george at math.northwestern.edu
More informationA STUDY OF q-lagranges POLYNOMIALS OF THREE VARIABLES
Pro Mathematica Vol. XVIII, Nos. 35-36, 24 A STUDY OF q-lagranges POLYNOMIALS OF THREE VARIABLES Mumtaz Ahmad Khan and Abdul Rahman Khan Abstract The present paper introduces a q-analogue of Lagranges
More informationAn Algebraic Identity of F.H. Jackson and its Implications for Partitions.
An Algebraic Identity of F.H. Jackson and its Implications for Partitions. George E. Andrews ( and Richard Lewis (2 ( Department of Mathematics, 28 McAllister Building, Pennsylvania State University, Pennsylvania
More informationOn q-series Identities Arising from Lecture Hall Partitions
On q-series Identities Arising from Lecture Hall Partitions George E. Andrews 1 Mathematics Department, The Pennsylvania State University, University Par, PA 16802, USA andrews@math.psu.edu Sylvie Corteel
More informationSingular Overpartitions
Singular Overpartitions George E. Andrews Dedicated to the memory of Paul Bateman and Heini Halberstam. Abstract The object in this paper is to present a general theorem for overpartitions analogous to
More informationSome New Partition Theorems
JOURNAL OF COMBINATORIAL THEORY 2, 431-436 (1967) Some New Partition Theorems GEORGE E. ANDREWS* Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania Communicated
More informationThe Concept of Bailey Chains
The Concept of Bailey Chains Peter Paule 0 Introduction In his expository lectures on q-series [3] G E Andrews devotes a whole chapter to Bailey s Lemma (Th 2, 3) and discusses some of its numerous possible
More informationBeukers integrals and Apéry s recurrences
2 3 47 6 23 Journal of Integer Sequences, Vol. 8 (25), Article 5.. Beukers integrals and Apéry s recurrences Lalit Jain Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G CANADA lkjain@uwaterloo.ca
More information2 J. ZENG THEOREM 1. In the ring of formal power series of x the following identities hold : (1:4) 1 + X n1 =1 S q [n; ]a x n = 1 ax? aq x 2 b x? +1 x
THE q-stirling NUMBERS CONTINUED FRACTIONS AND THE q-charlier AND q-laguerre POLYNOMIALS By Jiang ZENG Abstract. We give a simple proof of the continued fraction expansions of the ordinary generating functions
More informationFOUR IDENTITIES RELATED TO THIRD ORDER MOCK THETA FUNCTIONS IN RAMANUJAN S LOST NOTEBOOK HAMZA YESILYURT
FOUR IDENTITIES RELATED TO THIRD ORDER MOCK THETA FUNCTIONS IN RAMANUJAN S LOST NOTEBOOK HAMZA YESILYURT Abstract. We prove, for the first time, a series of four related identities from Ramanujan s lost
More informationNew modular relations for the Rogers Ramanujan type functions of order fifteen
Notes on Number Theory and Discrete Mathematics ISSN 532 Vol. 20, 204, No., 36 48 New modular relations for the Rogers Ramanujan type functions of order fifteen Chandrashekar Adiga and A. Vanitha Department
More informationElementary proofs of congruences for the cubic and overcubic partition functions
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 602) 204), Pages 9 97 Elementary proofs of congruences for the cubic and overcubic partition functions James A. Sellers Department of Mathematics Penn State
More informationCONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS
Bull. Aust. Math. Soc. 9 2016, 400 409 doi:10.1017/s000497271500167 CONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS M. S. MAHADEVA NAIKA, B. HEMANTHKUMAR H. S. SUMANTH BHARADWAJ Received 9 August
More informationMACMAHON S PARTITION ANALYSIS IX: k-gon PARTITIONS
MACMAHON S PARTITION ANALYSIS IX: -GON PARTITIONS GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Dedicated to George Szeeres on the occasion of his 90th birthday Abstract. MacMahon devoted a significant
More informationCONSTRUCTIVE APPROXIMATION
Constr. Approx. (1993) 9:509-523 CONSTRUCTIVE APPROXIMATION 9 1993 Springer-Verlag New York Inc. Hypergeometric Analogues of the Arithmetic-Geometric Mean Iteration J. Borwein, P. Borwein, F. Garvan Abstract.
More informationDOUBLE INTEGRAL REPRESENTATION AND CERTAIN TRANSFORMATIONS FOR BASIC APPELL FUNCTIONS
ISSN 2348-28 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-226 (online) Vol. 2, Issue 3, pp: (9-3), Month: July 24 - September 24, Available at: www.researchpublish.com
More informationAnalogues of Ramanujan s 24 squares formula
International Journal of Number Theory Vol., No. 5 (24) 99 9 c World Scientific Publishing Company DOI:.42/S79342457 Analogues of Ramanujan s 24 squares formula Faruk Uygul Department of Mathematics American
More informationA SIMPLE EVALUATION OF ASKEY AND WILSON'S? BETA INTEGRAL MIZAN RAHMAN1
proceedings of the american mathematical society Volume 92. Number 3, November 1984 A SIMPLE EVALUATION OF ASKEY AND WILSON'S? BETA INTEGRAL MIZAN RAHMAN1 Abstract. By using the well-known sum of 2
More informationarxiv: v3 [math.ca] 24 Sep 2013
Several special values of Jacobi theta functions arxiv:06.703v3 [math.ca] Sep 03 Abstract István Mező, Using the duplication formulas of the elliptic trigonometric functions of Gosper, we deduce some new
More informationAn identity of Andrews and the Askey-Wilson integral
Ramanujan J DOI 0.007/s39-008-922-4 An identity of Andrews and the Askey-Wilson integral Zhi-Guo Liu Received: 6 July 2007 / Accepted: 7 January 2008 Springer Science+Business Media, LLC 2008 Abstract
More informationSome generalizations of a supercongruence of van Hamme
Some generalizations of a supercongruence of van Hamme Victor J. W. Guo School of Mathematical Sciences, Huaiyin Normal University, Huai an, Jiangsu 3300, People s Republic of China jwguo@hytc.edu.cn Abstract.
More informationCOMBINATORICS OF GENERALIZED q-euler NUMBERS. 1. Introduction The Euler numbers E n are the integers defined by E n x n = sec x + tan x. (1.1) n!
COMBINATORICS OF GENERALIZED q-euler NUMBERS TIM HUBER AND AE JA YEE Abstract New enumerating functions for the Euler numbers are considered Several of the relevant generating functions appear in connection
More informationNew Congruences for Broken k-diamond Partitions
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018), Article 18.5.8 New Congruences for Broken k-diamond Partitions Dazhao Tang College of Mathematics and Statistics Huxi Campus Chongqing University
More informationarxiv: v2 [math.nt] 9 Apr 2015
CONGRUENCES FOR PARTITION PAIRS WITH CONDITIONS arxiv:408506v2 mathnt 9 Apr 205 CHRIS JENNINGS-SHAFFER Abstract We prove congruences for the number of partition pairs π,π 2 such that π is nonempty, sπ
More informationTHE At AND Q BAILEY TRANSFORM AND LEMMA
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 2, April 1992 THE At AND Q BAILEY TRANSFORM AND LEMMA STEPHEN C. MILNE AND GLENN M. LILLY Abstract. We announce a higher-dimensional
More informationCongruence Properties of Partition Function
CHAPTER H Congruence Properties of Partition Function Congruence properties of p(n), the number of partitions of n, were first discovered by Ramanujan on examining the table of the first 200 values of
More informationGenerating Functions of Partitions
CHAPTER B Generating Functions of Partitions For a complex sequence {α n n 0,, 2, }, its generating function with a complex variable q is defined by A(q) : α n q n α n [q n ] A(q). When the sequence has
More informationA GENERALIZATION OF THE FARKAS AND KRA PARTITION THEOREM FOR MODULUS 7
A GENERALIZATION OF THE FARKAS AND KRA PARTITION THEOREM FOR MODULUS 7 S. OLE WARNAAR Dedicated to George Andrews on the occasion of his 65th birthday Abstract. We prove generalizations of some partition
More informationEQUIVALENCE OF THE CLASSICAL THEOREMS OF SCHOTTKY, LANDAU, PICARD AND HYPERBOLICITY
proceedings of the american mathematical society Volume 89, Number 4. December 1983 EQUIVALENCE OF THE CLASSICAL THEOREMS OF SCHOTTKY, LANDAU, PICARD AND HYPERBOLICITY KY0NGT. HAHN1 Abstract. Modifying
More informationInteger Partitions and Why Counting Them is Hard
Portland State University PDXScholar University Honors Theses University Honors College 3-3-207 Integer Partitions and Why Counting Them is Hard Jose A. Ortiz Portland State University Let us know how
More informationAn Interesting q-continued Fractions of Ramanujan
Palestine Journal of Mathematics Vol. 4(1 (015, 198 05 Palestine Polytechnic University-PPU 015 An Interesting q-continued Fractions of Ramanujan S. N. Fathima, T. Kathiravan Yudhisthira Jamudulia Communicated
More information2-UNIVERSAL POSITIVE DEFINITE INTEGRAL QUINARY QUADRATIC FORMS
2-UNIVERSAL POSITIVE DEFINITE INTEGRAL QUINARY QUADRATIC FORMS Byeong Moon Kim, Myung-Hwan Kim and Byeong-Kweon Oh Dept. of Math., Kangnung Nat l Univ., Kangwondo 210-702, Korea (kbm@knusun.kangnung.ac.kr)
More informationJacobi s Two-Square Theorem and Related Identities
THE RAMANUJAN JOURNAL 3, 153 158 (1999 c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Jacobi s Two-Square Theorem and Related Identities MICHAEL D. HIRSCHHORN School of Mathematics,
More informationArithmetic Relations for Overpartitions
Arithmetic Relations for Overpartitions Michael D. Hirschhorn School of Mathematics, UNSW, Sydney 2052, Australia m.hirschhorn@unsw.edu.au James A. Sellers Department of Mathematics The Pennsylvania State
More informationIDENTITIES FOR OVERPARTITIONS WITH EVEN SMALLEST PARTS
IDENTITIES FOR OVERPARTITIONS WITH EVEN SMALLEST PARTS MIN-JOO JANG AND JEREMY LOVEJOY Abstract. We prove several combinatorial identities involving overpartitions whose smallest parts are even. These
More informationON RECENT CONGRUENCE RESULTS OF ANDREWS AND PAULE FOR BROKEN ifc-diamonds
BULL. AUSTRAL. MATH. SOC. VOL. 75 (2007) [121-126] 05A17, 11P83 ON RECENT CONGRUENCE RESULTS OF ANDREWS AND PAULE FOR BROKEN ifc-diamonds MICHAEL D. HIRSCHHORN AND JAMES A. SELLERS In one of their most
More information