An Interesting q-continued Fractions of Ramanujan
|
|
- Ursula Gordon
- 5 years ago
- Views:
Transcription
1 Palestine Journal of Mathematics Vol. 4(1 (015, Palestine Polytechnic University-PPU 015 An Interesting q-continued Fractions of Ramanujan S. N. Fathima, T. Kathiravan Yudhisthira Jamudulia Communicated by Zafar Ahsan MSC 010 Classifications: 11A55. Keywords phrases: Continued fractions, Modular Equations. The first-named author is thankful to UGC, New Delhi for awarding research project [No. F41-139/01/(SR], under which this work has been done. Abstract. In this paper, we establish an interesting q-identity an integral representation of a q-continued fraction of Ramanujan. We also compute explicit evaluation of this continued fraction derive its relation with Ramanujan Göllnitz -Gordon continued fraction. 1 Introduction Ramanujan a pioneer in the theory of continued fraction has recorded several in the process rediscovered few continued fractions found earlier by Gauss, Eisenstein Rogers in his notebook [10]. In fact Chapter 1 Chapter 16 of his Second Notebook [10] is devoted to continued fractions. Proofs of these continued fractions over years are given by several mathematician, we mention here specially G.E. Andrews[3], C. Adiga, S. Bhargava G.N. Watson [1] whose works have been compiled in [4] [5]. The celebrated Roger Ramanujan continued fraction is defined by R(q := 1 q q q 3, q < 1. On page 365 of his lost notebook [11], Ramanujan recorded five modular equations relating R(q with R( q, R(q, R(q 3 R(q 4 R(q 5. The well known Ramanujan s cubic continued fraction defined by J(q := q1/3 q + q q + q 4 q 3 + q 6, q < 1. is recorded on page 366 of his lost notebook [11]. Several new modular equation relating J(q with J( q, J(q J(q 3 are established by H.H. Chan [8]. Similarly the Ramanujan Göllnitz-Gordon continued fraction K(q defined by K(q := q1/ q + q q 3 + q 4 q 5 + q 6 q 7 +, q < 1, satisfies several beautiful modular relations. One may see traces of modular equation related to K(q on page 9 of Ramanujan s lost notebook [11]. Further works related to K(q in recent years have been done by various authors including Chan S.S Huang [9] K.R. Vasuki B.R. Srivatsa Kumar [1]. Motivated by these works in this paper we study the Ramanujan continued fraction M(q := q1/ 1 q + q(1 q (1 q( q + q(1 q 3 (1 q( q 4 + q(1 q 5 (1 q( q 6 +, q < 1 = q 1/ (q4 ; q 4 (q ; q 4. (1.1 In Chapter 16 Entry 1 of [5], Ramanujan has recorded the following continued fraction (a q 3 ; q 4 (b q 3 ; q 4 (a q; q 4 (b q; q 4 = 1 1 ab + (a bq(b aq (1 ab( q + (a bq 3 (b aq 3 (1 ab( q 4 +, ab < 1, q < 1. (1.
2 An Interesting q-continued Fractions of Ramanujan 199 Ra- In fact setting a = q 1/ b = q 1/ in (1., we obtain (1.1. In Section we obtain an interesting q-identity related to M(q using manujan s 1 ψ 1 summation formula [5, Ch. 16, Entry 17] n= Andrew s identity [4, p. 57], (a n (b n z n = (az (q/az (q (b/a (z (b/az (b (q/a, b/a < z < 1, (1.3 q kn 1 q = ln+k q ln +kn qln+k. (1.4 1 qln+k In Section 3 we obtain several relation of M(q with theta function ϕ(q, ψ(q χ(q. In Section 4 we obtain an integral representation of M(q. In Section 5 we derive a formula that help us to obtain relation among M(q 1/, M(q, M(q M(q 4. We establish explicit formulas for the evaluation of M(e πn in Section 6. We conclude this introduction with few customary definition we make use in the sequel. For a q complex number with q < 1 (a := (a; q = (a n := (a; q n = (1 aq n n 1 (1 aq k = (a (aq n, k=0 n : any integer. f(a, b := n= a n(n+1/ b n(n 1/ (1.5 = ( a; ab ( b; ab (ab; ab, ab < 1. (1.6 Identity (1.6 is the Jacobi s triple product identity in Ramanujan s notation 16, Entry 19]. It follows from (1.5 (1.6 that [5, Ch. 16, Entry ], [5, Ch. ϕ(q := f(q, q = n= q n = ( q; q (q; q, (1.7 ψ(q := f(q, q 3 = q n(n+1/ = (q ; q (q; q, (1.8 χ(q := ( q; q. (1.9 q-identity related to M(q Theorem.1 M(q = n(8n+4+1/ q8n+ q 1 q 8n+ (n+1(8n+4+1/ q8n+6 q 1 q 8n+6 (.1 Proof: Changing q to q 8, then setting a = q, b = q 10 z = q in 1 ψ 1 summation formula (1.3 we obtain (q 4 ; q 4 (q ; q 4 = q n 1 q 8n+ q 6n+4 1 q 8n+6 (.
3 00 S. N. Fathima, T. Kathiravan Yudhisthira Jamudulia employing Andrews identity (1.4 with k =, l = 8 k = 6, l = 8 in both the summations in right side of the identity (. respectively finally multiplying both sides of the resulting identity with q 1/ using product represtation of M(q (1.1, we complete the proof of Theorem.1. 3 Some Identities involving M(q We obtain relation of M(q in terms of theta function ϕ(q, ψ(q χ(q. Theorem 3.1 M(q = q 1/ ψ4 (q ϕ (q, (3.1 8M(q = ϕ (q ϕ ( q, (3. 16M (q = ϕ 4 (q ϕ 4 ( q, (3.3 M (q M(q M 1 (q + M(q M 1 (q M(q = ϕ (q, (3.4 4M(q = ϕ (q ϕ (q, (3.5 = qψ4 (q 1 qψ 4 (q, (3.6 8M(q = χ (q χ ( q φ ( q φ ( q. (3.7 Proof: Using [5, Ch. 16, Entry (ii] in (1.1, we obtain M(q = q 1/ ψ (q. (3.8 Employing [5, Ch. 16, Entry 5(iv] in (3.8, we obtain (3.1. From (3.1 we have M(q = q ψ4 (q ϕ (q. (3.9 Employing [5, Ch. 16, Entry 5(vii] [5, Ch. 16, Entry 5(vi] in (3.9, we obtain (3.. Identity (3.3 immediately follows from (3.8 [5, Ch. 16, Entry 5(vii]. Again from (3.1, we have M (q M(q = ψ8 (qϕ (q ψ 4 (q ϕ 4 (q, (3.10 employing [5, Ch. 16, Entry 5(iv] in the identity (3.10 we obtain (3.4. From (3. (3.3, we have 64M (q + 16M (q = 16ϕ (qm(q, (3.11 dividing the identity (3.11 throughout by 16M(q using (3.4 we obtain (3.5. From (3.1 we deduce that M 1 (q + M(q = ϕ4 (q + q ψ 8 (q q 1/ ϕ (qψ 4 (q, (3.1 M 1 (q M(q = ϕ4 (q q ψ 8 (q q 1/ ϕ (qψ 4 (q. (3.13 On dividing (3.1 by (3.13 using [5, Ch. 16, Entry 5(iv] in the resulting identity, we complete the proof of (3.6. From (1.7 (1.9 we have ϕ( q + χ(q χ( q ϕ( q = (q; q [ ( q; q ] f(q, q, f( q, q
4 An Interesting q-continued Fractions of Ramanujan 01 employing [5, Ch. 16, Entry 30(ii] in right h side of above identity we obtain ϕ( q + χ(q χ( q ϕ( q = (q 8 ; q 8 5 (q 3 ; q 3 (q 4 ; q 4 (q 16 ; q 16 (q 64 ; q 64 4 M(q 16 q 8. (3.14 Again from (1.7 (1.9 we have ϕ( q χ(q χ( q ϕ( q = (q; q [ 1 ( q; q ] f(q, q, f( q, q employing [5, Ch. 16, Entry 30(ii] [5, Ch. 16, Entry 18(ii] in right h side of above identity we obtain ϕ( q χ(q χ( q ϕ( q = 4q( q8 ; q 8 (q 64 ; q 64 3 q 8 ( q 16 ; q 3 M(q 16. (3.15 Multiplying (3.14 (3.15 we complete the proof of (3.7. Theorem 3.. Let u = M(q, v = M( q w = M(q, then u v = 8w Proof: On substituting (3.4 in (3.5, we obtain ϕ (q = 4M (q + M (q M(q Changing q to q in (3.16, we have ϕ ( q = 4M (q + M ( q M(q. (3.16. (3.17 Subtracting (3.17 from (3.16 using identity (3., we complete the proof of Theorem Integral Representation of M(q Theorem 4.1. For 0 < q < 1, ( 1 M(q = exp q + 4 [ ϕ 4 ( q 1 q 8 where ϕ(q ψ(q are as defined in (1.7 (1.8. Proof: Taking log on both sides of (3.1, we have + q ] ϕ (q dq, (4.1 ϕ(q log M(q = 1 log q + 4 log ψ(q log ϕ(q. (4. Employing [5, Ch. 16, Entry 3(ii] [5, Ch. 16, Entry 3(i] on right h side of (4., we obtain log M(q = 1 log q + 4 q n n( q n. (4.3 n=1 Differentiating (4.3 simplifying, we have d dq log M(q = 1 q + 4 [ q n=1 ( 1 n q n ( q n + n=1 ] q n 1 ( q n 1. (4.4 Using Jacobi s identity [5, Ch. 16, Identity 33.5, p. 54] [5, Ch. 16, Entry 3(i] integrating both sides finally exponentiating both sides of identity (4.4, we complete the proof of Theorem 4.1.
5 0 S. N. Fathima, T. Kathiravan Yudhisthira Jamudulia 5 Modular Equation of Degree n Relation Between M(q M(q n In the terminology of hypergeometric function, a modular equation of degree n is a relation between α β that is induced by where n F 1 (1/, 1/; 1; 1 α F 1 (1/, 1/; 1; α F 1 (a, b; c; x = (a k = k=0 = F 1 (1/, 1/; 1; 1 β F 1 (1/, 1/; 1; β (a k (b k (c k k! Γ(a + k Γ(a. x k,, Let Z 1 (r = F 1 (1/r, r 1/r; 1; α Z n (r = F 1 (1/r, r 1/r; 1; β, where n is the degree of the modular equation. The multiplier m(r is defined by the equation Theorem 5.1. If then m(r = Z 1(r Z n (r. ( q = exp α = 16 M 4 (q M 4 (q 1/ Proof: From (1.1 (1.7, we have M(qϕ (q = q 1/ (q4 ; q 4 (q ; q 4 π F 1 (1/, 1/; 1; 1 α F 1 (1/, 1/; 1; α ( q; q 4 (q 4 ; q 4 ( q ; q 4 (q ; q 4, (5.1 = M (q 1/. (5.3 Substituting (5.3 in (3.3, we obtain 16M (q = M 4 (q 1/ [ M (q 1 ϕ4 ( q ϕ 4 (q ]. (5.4 From a known identity [5, Ch. 16, p. 100, Entry 5] (5.1 it is implied that α = 1 ϕ4 ( q ϕ 4 (q. (5.5 Using (5.5 in (5.4, we complete the proof of (5.. Let α β be related by (5.1. If β has degree n over α then from Theorem 5.1, we obtain β = 16 M 4 (q n M 4 (q n/. (5.6 (5. Corollary 5.. Let u = M(q 1/, v = M(q, w = M(q x = M(q 4, then 16x 4 v + 3x 3 wv 4x 3 wu 4 + 4x w v + 8xw 3 v xw 3 u 4 + w 4 v = 0. (5.7 Proof: From [5, Entry 4(v, p. 16], we have ( 1 β 1/4 1 α = β 1/4. (5.8
6 An Interesting q-continued Fractions of Ramanujan 03 On using (5.6 with n = 4 (5. in (5.8, we obtain u 4 16v 4 ( w x u 4 =. (5.9 w + x Squaring both side of (5.9 then simplifying, we obtain ( Evaluations of M(q As an application of Theorem 5.1, we establish few explicit evaluation of M(q. Let q n = e πn let α n denote the corresponding value of α in (5.1. Then by Theorem 5.1, we have M(e πn = 1 α1/4 n (6.1 From [5, Ch. 17, p. 97], we have α 1 = 1, α = ( 1 α 4 = ( 1 4. Thus from (6.1, it immediately follows M(e π M(e π/ M(e π M(e π/ M(e π M(e π = ( 5/4 1, (6. = 1 1, (6.3 = 1. (6.4 Ramanujan has recorded several modular equation in his notebook [10, p ] [10, p ] which are very useful in the computation of class invariants the values of theta function. Ramanujan has also recorded several values of theta function ϕ(q ψ(q in his notebook. For example ϕ(e π = π 1/4 Γ(3/4, (6.5 ψ(e π = 5/8 e π/8 π 1/4 ϕ(e π ϕ(e 3π From (3.8 (6.6, we have M(e π/ = 5/4 Γ(3/4, (6.6 = (6.7 π Γ (3/4, (6.8 Using (6.8 (6., we obtain π M(e π = Γ (3/. (6.9 Setting (6.9 in (6.4, we obtain 1 M(e π = π Γ (3/. (6.10 J.M. Borwein P.B. Borwein [7] are the first to observe that class invariant could be used
7 04 S. N. Fathima, T. Kathiravan Yudhisthira Jamudulia to evaluated certain values of ϕ(e nπ. The Ramanujan Weber class invariants are defined by G n := 1/4 qn 1/4 ( q n ; qn g n := 1/4 q 1/4 n (q n ; q n, (6.11 where q n = e πn. Chan Huang has derived few explicit formulas for evaluating K(e πn/ in the terms of Ramanujan Weber class. Similar works are done by Adiga et., al. Analogoues to n these works we obtain explicit formulas to evaluate M(e π. Theorem 6.1. For Ramanujan Weber class invariant defined as in (6.11, let p = G 1 n p 1 = gn 1, then M(e πn M(e πn = 1 1 p (p p (p 1, (6.1 = 1 p 1 p 1. (6.13 Proof: From [9], we have Hence G n = [4α n (1 α n ] 1/4. α n = Using (6.14 in (6.1, we obtain (6.1. Also from [9], we have Hence 1 ( p(p p(p 1. (6.14 g 1 n = 1 αn α n. αn = (p 1 p 1. (6.15 Using (6.15 in (6.1, we complete the proof of (6.13. Example: Let n = 1. Since G 1 = 1, from Theorem 6.1 we have M(e π M(e π/ = ( 5/4 1. Let n =. Since g = 1, from Theorem 6.1 we have M(e π M(e π/ = 1 1. Remark: Using [10, p. 9] it is easily verified that M(q K(q are related by the equation M(q K(q + K(qM(q M(q = 0. References [1] C. Adiga, B. C. Berndt, S.Bhargava G. N. Watson, Chapter 16 of Ramanujan s Second Notebook: Theta-Functions q-series, Mem. Amer. Math. Soc., No. 315, 53(1985, 1-85, Amer. Math. Soc., Providence, [] C. Adiga, T. Kim, M. S. Mahadeva Naika H. S. Madhusudhan, On Ramanujan s Cubic Continued Fraction Explicit Evaluations of Theta Functions,Indian J. Pure App. Math., No. 55, 9(004,
8 An Interesting q-continued Fractions of Ramanujan 05 [3] G. E. Andrews, An introduction to Ramanujan s lost notebook, Amer. Math. Monthly, 86(1979, [4] G. E. Andrews, The Lost Notebook Part I, Springer Verlang, New York, 005. [5] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York, [6] B. C. Berndt, Ramanujan s Notebooks, Part V, Springer-Verlag, New York, [7] J. M. Borwein P. B. Borwein, Pi the AGM, Wiley, New York, [8] H.H. Chan, On Ramanujan s Cubic Continued Fraction, Acta Arith. 73(1995, [9] H.H. Chan S.S. Huang, On the Ramanujan Göllnitz Gordon Continued Fraction, Ramanujan J., 1(1997, [10] S. Ramanujan, Notebooks, volumes, Tata Institute of Fundamental Research, Bombay, [11] S. Ramanujan, The Lost Notebook Other Unpublished Papers, Narosa, New Delhi, [1] K.R. Vasuki B.R. Srivatsa Kumar, Certain identities for the Ramanujan Gollnitz Gordon Continued fraction, J. of Comp. Applied Mathematics. 187(006, Author information S. N. Fathima, T. Kathiravan Yudhisthira Jamudulia, Department of Mathematics, Ramanujan School of Mathematics, Pondicherry University, Puducherry , INDIA. fathima.mat@pondiuni.edu.in; kkathiravan98@gmail.com; yudhis4581@yahoo.co.in Received: October 7, 013. Accepted: January 11, 014.
ON A CONTINUED FRACTION IDENTITY FROM RAMANUJAN S NOTEBOOK
Asian Journal of Current Engineering and Maths 3: (04) 39-399. Contents lists available at www.innovativejournal.in ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage: http://www.innovativejournal.in/index.php/ajcem
More informationM. S. Mahadeva Naika, S. Chandankumar and N. P. Suman (Received August 6, 2012)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 42 2012, 157-175 CERTAIN NEW MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION M. S. Mahadeva Naika, S. Chandankumar and N.. Suman Received August 6,
More informationMODULAR EQUATIONS FOR THE RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 40 010, -48 MODULAR EUATIONS FOR THE RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR Received August
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 informationCertain Somos s P Q type Dedekind η-function identities
roc. Indian Acad. Sci. (Math. Sci.) (018) 18:4 https://doi.org/10.1007/s1044-018-04-3 Certain Somos s Q type Dedekind η-function identities B R SRIVATSA KUMAR H C VIDYA Department of Mathematics, Manipal
More informationSOME CONGRUENCES FOR PARTITION FUNCTIONS RELATED TO MOCK THETA FUNCTIONS ω(q) AND ν(q) S.N. Fathima and Utpal Pore (Received October 13, 2017)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 47 2017), 161-168 SOME CONGRUENCES FOR PARTITION FUNCTIONS RELATED TO MOCK THETA FUNCTIONS ωq) AND νq) S.N. Fathima and Utpal Pore Received October 1, 2017) Abstract.
More informationJournal of Number Theory
Journal of Number Theory 133 2013) 437 445 Contents lists available at SciVerse ScienceDirect Journal of Number Theory wwwelseviercom/locate/jnt On Ramanujan s modular equations of degree 21 KR Vasuki
More information( 1) n q n(3n 1)/2 = (q; q). (1.3)
MATEMATIQKI VESNIK 66, 3 2014, 283 293 September 2014 originalni nauqni rad research paper ON SOME NEW MIXED MODULAR EQUATIONS INVOLVING RAMANUJAN S THETA-FUNCTIONS M. S. Mahadeva Naika, S. Chandankumar
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 informationRamanujan s modular equations and Weber Ramanujan class invariants G n and g n
Bull. Math. Sci. 202) 2:205 223 DOI 0.007/s3373-0-005-2 Ramanujan s modular equations and Weber Ramanujan class invariants G n and g n Nipen Saikia Received: 20 August 20 / Accepted: 0 November 20 / Published
More informationJournal of Mathematical Analysis and Applications
J Math Anal Appl 396 (0) 3 0 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis Applications journal homepage: wwwelseviercom/locate/jmaa On Ramanujan s modular equations
More informationNew congruences for overcubic partition pairs
New congruences for overcubic partition pairs M. S. Mahadeva Naika C. Shivashankar Department of Mathematics, Bangalore University, Central College Campus, Bangalore-560 00, Karnataka, India Department
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 informationOn the expansion of Ramanujan's continued fraction of order sixteen
Tamsui Oxford Journal of Information and Mathematical Sciences 31(1) (2017) 81-99 Aletheia University On the expansion of Ramanujan's continued fraction of order sixteen A. Vanitha y Department of Mathematics,
More informationSOME MODULAR EQUATIONS IN THE FORM OF SCHLÄFLI 1
italian journal of pure and applied mathematics n. 0 01 5) SOME MODULAR EQUATIONS IN THE FORM OF SCHLÄFLI 1 M.S. Mahadeva Naika Department of Mathematics Bangalore University Central College Campus Bengaluru
More informationResearch Article Some New Explicit Values of Quotients of Ramanujan s Theta Functions and Continued Fractions
International Mathematics and Mathematical Sciences Article ID 534376 7 pages http://dx.doi.org/0.55/04/534376 Research Article Some New Explicit Values of Quotients of Ramanujan s Theta Functions and
More informationPARTITION IDENTITIES AND RAMANUJAN S MODULAR EQUATIONS
PARTITION IDENTITIES AND RAMANUJAN S MODULAR EQUATIONS NAYANDEEP DEKA BARUAH 1 and BRUCE C. BERNDT 2 Abstract. We show that certain modular equations and theta function identities of Ramanujan imply elegant
More informationBruce C. Berndt, Heng Huat Chan, and Liang Cheng Zhang. 1. Introduction
RADICALS AND UNITS IN RAMANUJAN S WORK Bruce C. Berndt, Heng Huat Chan, and Liang Cheng Zhang In memory of S. Chowla. Introduction In problems he submitted to the Journal of the Indian Mathematical Society
More informationArithmetic properties of overcubic partition pairs
Arithmetic properties of overcubic partition pairs Bernard L.S. Lin School of Sciences Jimei University Xiamen 3101, P.R. China linlsjmu@13.com Submitted: May 5, 014; Accepted: Aug 7, 014; Published: Sep
More informationON 2- AND 4-DISSECTIONS FOR SOME INFINITE PRODUCTS ERNEST X.W. XIA AND X.M. YAO
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 6, 2013 ON 2- AND 4-DISSECTIONS FOR SOME INFINITE PRODUCTS ERNEST X.W. XIA AND X.M. YAO ABSTRACT. The 2- and 4-dissections of some infinite products
More informationSOME THETA FUNCTION IDENTITIES RELATED TO THE ROGERS-RAMANUJAN CONTINUED FRACTION
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2895 2902 S 0002-99399804516-X SOME THETA FUNCTION IDENTITIES RELATED TO THE ROGERS-RAMANUJAN CONTINUED FRACTION
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 informationPartition identities and Ramanujan s modular equations
Journal of Combinatorial Theory, Series A 114 2007 1024 1045 www.elsevier.com/locate/jcta Partition identities and Ramanujan s modular equations Nayandeep Deka Baruah 1, Bruce C. Berndt 2 Department of
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 informationOn Continued Fraction of Order Twelve
Pue Mathematical Sciences, Vol. 1, 2012, no. 4, 197-205 On Continued Faction of Ode Twelve B. N. Dhamenda*, M. R. Rajesh Kanna* and R. Jagadeesh** *Post Gaduate Depatment of Mathematics Mahaani s Science
More informationFLOWERS WHICH WE CANNOT YET SEE GROWING IN RAMANUJAN S GARDEN OF HYPERGEOMETRIC SERIES, ELLIPTIC FUNCTIONS, AND q S
FLOWERS WHICH WE CANNOT YET SEE GROWING IN RAMANUJAN S GARDEN OF HYPERGEOMETRIC SERIES, ELLIPTIC FUNCTIONS, AND q S BRUCE C. BERNDT Abstract. Many of Ramanujan s ideas and theorems form the seeds of questions
More informationResearch Article A Parameter for Ramanujan s Function χ(q): Its Explicit Values and Applications
International Scholarly Research Network ISRN Computational Mathematics Volume 01 Article ID 169050 1 pages doi:1050/01/169050 Research Article A Parameter for Ramanujan s Function χq: Its Explicit Values
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 informationSOME THEOREMS ON THE ROGERS RAMANUJAN CONTINUED FRACTION IN RAMANUJAN S LOST NOTEBOOK
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 5, Pages 257 277 S 0002-9947(00)02337-0 Article electronically published on February 8, 2000 SOME THEOREMS ON THE ROGERS RAMANUJAN CONTINUED
More information2011 Boonrod Yuttanan
0 Boonrod Yuttanan MODULAR EQUATIONS AND RAMANUJAN S CUBIC AND QUARTIC THEORIES OF THETA FUNCTIONS BY BOONROD YUTTANAN DISSERTATION Submitted in partial fulfillment of the requirements for the degree of
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 informationRECURRENCE RELATION FOR COMPUTING A BIPARTITION FUNCTION
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 48, Number, 08 RECURRENCE RELATION FOR COMPUTING A BIPARTITION FUNCTION D.S. GIREESH AND M.S. MAHADEVA NAIKA ABSTRACT. Recently, Merca [4] found the recurrence
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics ELLIPTIC FUNCTIONS TO THE QUINTIC BASE HENG HUAT CHAN AND ZHI-GUO LIU Volume 226 No. July 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 226, No., 2006 ELLIPTIC FUNCTIONS TO THE
More informationINCOMPLETE ELLIPTIC INTEGRALS IN RAMANUJAN S LOST NOTEBOOK. 1. Introduction
INCOMPLETE ELLIPTIC INTEGRALS IN RAMANUJAN S LOST NOTEBOOK BRUCE C. BERNDT, HENG HUAT CHAN, AND SEN SHAN HUANG Dedicated with appreciation thanks to Richard Askey on his 65 th birthday Abstract. On pages
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics A CERTAIN QUOTIENT OF ETA-FUNCTIONS FOUND IN RAMANUJAN S LOST NOTEBOOK Bruce C Berndt Heng Huat Chan Soon-Yi Kang and Liang-Cheng Zhang Volume 0 No February 00 PACIFIC JOURNAL
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 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 congruences for Andrews Paule s broken 2-diamond partitions
Discrete Mathematics 308 (2008) 5735 5741 www.elsevier.com/locate/disc Some congruences for Andrews Paule s broken 2-diamond partitions Song Heng Chan Division of Mathematical Sciences, School of Physical
More informationCongruences modulo 3 for two interesting partitions arising from two theta function identities
Note di Matematica ISSN 113-53, e-issn 1590-093 Note Mat. 3 01 no., 1 7. doi:10.185/i1590093v3n1 Congruences modulo 3 for two interesting artitions arising from two theta function identities Kuwali Das
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 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 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 informationA New Form of the Quintuple Product Identity and its Application
Filomat 31:7 (2017), 1869 1873 DOI 10.2298/FIL1707869S Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A New Form of the Quintuple
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: v1 [math.co] 8 Sep 2017
NEW CONGRUENCES FOR BROKEN k-diamond PARTITIONS DAZHAO TANG arxiv:170902584v1 [mathco] 8 Sep 2017 Abstract The notion of broken k-diamond partitions was introduced by Andrews and Paule Let k (n) denote
More informationResearch Article Some Congruence Properties of a Restricted Bipartition
International Analysis Volume 06, Article ID 90769, 7 pages http://dx.doi.org/0.55/06/90769 Research Article Some Congruence Properties of a Restricted Bipartition Function c N (n) Nipen Saikia and Chayanika
More informationRamanujan and the Modular j-invariant
Canad. Math. Bull. Vol. 4 4), 1999 pp. 47 440 Ramanujan and the Modular j-invariant Bruce C. Berndt and Heng Huat Chan Abstract. A new infinite product t n was introduced by S. Ramanujan on the last page
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 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 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 informationarxiv:math/ v2 [math.qa] 22 Jun 2006
arxiv:math/0601181v [mathqa] Jun 006 FACTORIZATION OF ALTERNATING SUMS OF VIRASORO CHARACTERS E MUKHIN Abstract G Andrews proved that if n is a prime number then the coefficients a k and a k+n of the product
More informationResearch Article On a New Summation Formula for
International Mathematics and Mathematical Sciences Volume 2011, Article ID 132081, 7 pages doi:10.1155/2011/132081 Research Article On a New Summation Formula for 2ψ 2 Basic Bilateral Hypergeometric Series
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 informationA Remarkable Cubic Mean Iteration
- Computational Methods Function Theory Proceedings, Valparafso 1989 St. Ruseheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 145, pp. 27-1 (~) Springer Berlin Heidelberg
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 informationEulerian series in q-series and modular forms. Youn Seo Choi
Eulerian series in q-series and modular forms Youn Seo Choi abstrat Eulerian series is very interesting power series even though we do not know any single method to handle the general Eulerian series However,
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 informationFACULTY PROFILE. Research Areas : Special Functions, Number Theory, Partition Theory, Discrete Mathematics and Graph Theory.
FACULTY PROFILE 1. Name : Dr. Chandrashekara Adiga 2. Designation : Professor 3. Qualification : M.Sc. Ph.D. 4. Area of Specialization: Special Functions, Number Theory, Partition Theory, Discrete Mathematics
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 informationArithmetic Properties for Ramanujan s φ function
Arithmetic Properties for Ramanujan s φ function Ernest X.W. Xia Jiangsu University ernestxwxia@163.com Nankai University Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function
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 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(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 informationAPPLICATIONS OF THE HEINE AND BAUER-MUIR TRANSFORMATIONS TO ROGERS-RAMANUJAN TYPE CONTINUED FRACTIONS
APPLICATIONS OF THE HEINE AND BAUER-MUIR TRANSFORMATIONS TO ROGERS-RAMANUJAN TYPE CONTINUED FRACTIONS JONGSIL LEE, JAMES MC LAUGHLIN AND JAEBUM SOHN Abstract. In this paper we show that various continued
More information(6n + 1)( 1 2 )3 n (n!) 3 4 n, He then remarks that There are corresponding theories in which q is replaced by one or other of the functions
RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES Bruce C Berndt, S Bhargava, Frank G Garvan Contents Introduction Ramanujan s Cubic Transformation, the Borweins Cubic Theta Function Identity,
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 informationRAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7. D. Ranganatha
Indian J. Pure Appl. Math., 83: 9-65, September 07 c Indian National Science Academy DOI: 0.007/s36-07-037- RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 D. Ranganatha Department of Studies in Mathematics,
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 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 informationAdvances in Applied Mathematics 48(2012), Constructing x 2 for primes p = ax 2 + by 2
Advances in Applied Mathematics 4(01, 106-10 Constructing x for primes p ax + by Zhi-Hong Sun School of Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 3001, P.R. China E-mail: zhihongsun@yahoo.com
More informationGENERAL FAMILY OF CONGRUENCES MODULO LARGE POWERS OF 3 FOR CUBIC PARTITION PAIRS. D. S. Gireesh and M. S. Mahadeva Naika
NEW ZEALAND JOURNAL OF MATEMATICS Volume 7 017, 3-56 GENERAL FAMILY OF CONGRUENCES MODULO LARGE POWERS OF 3 FOR CUBIC PARTITION PAIRS D. S. Gireesh and M. S. Mahadeva Naika Received May 5, 017 Abstract.
More informationUNIFICATION OF MODULAR TRANSFORMATIONS FOR CUBIC THETA FUNCTIONS. (Received April 2003) 67vt
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), 2-27 UNIFICATION OF MODULAR TRANSFORMATIONS FOR CUBIC THETA FUNCTIONS S. B hargava Sye d a N oor Fathim a (Received April 2003) Abstract. We obtain
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 informationThe Value of the Zeta Function at an Odd Argument
International Journal of Mathematics and Computer Science, 4(009), no., 0 The Value of the Zeta Function at an Odd Argument M CS Badih Ghusayni Department of Mathematics Faculty of Science- Lebanese University
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 Klein s quartic curve
114 On Klein s quartic curve Maurice Craig 1 Introduction Berndt and Zhang [2] have drawn attention to three formulae at the foot of page 300 in Ramanujan s second notebook [17]. Of these formulae, the
More informationELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES
Bull. Aust. Math. Soc. 79 (2009, 507 512 doi:10.1017/s0004972709000136 ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES MICHAEL D. HIRSCHHORN and JAMES A. SELLERS (Received 18 September 2008 Abstract Using
More informationQUANTUM MODULARITY OF MOCK THETA FUNCTIONS OF ORDER 2. Soon-Yi Kang
Korean J. Math. 25 (2017) No. 1 pp. 87 97 https://doi.org/10.11568/kjm.2017.25.1.87 QUANTUM MODULARITY OF MOCK THETA FUNCTIONS OF ORDER 2 Soon-Yi Kang Abstract. In [9] we computed shadows of the second
More informationOn Certain Hypergeometric Summation Theorems Motivated by the Works of Ramanujan, Chudnovsky and Borwein
www.ccsenet.org/jmr Journal of Mathematics Research Vol., No. August 00 On Certain Hypergeometric Summation Theorems Motivated by the Works of Ramanujan, Chudnovsky and Borwein M. I. Qureshi Department
More informationCranks in Ramanujan s Lost Notebook
Cranks in Ramanujan s Lost Notebook Manjil P. Saikia Department of Mathematical Sciences, Tezpur University, Napaam Dist. - Sonitpur, Pin - 784028 India manjil@gonitsora.com January 22, 2014 Abstract We
More informationNOTES ON RAMANUJAN'S SINGULAR MODULI. Bruce C. Berndt and Heng Huat Chan. 1. Introduction
NOTES ON RAMANUJAN'S SINGULAR MODULI Bruce C. Berndt Heng Huat Chan 1. Introduction Singular moduli arise in the calculation of class invariants, so we rst dene the class invariants of Ramanujan Weber.
More informationMOCK THETA FUNCTIONS OF ORDER 2 AND THEIR SHADOW COMPUTATIONS
MOCK THETA FUNCTIONS OF ORDER AND THEIR SHADOW COMPUTATIONS SOON-YI KANG AND HOLLY SWISHER Abstract Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular
More informationRamanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3
Ramanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3 William Y.C. Chen 1, Anna R.B. Fan 2 and Rebecca T. Yu 3 1,2,3 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071,
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 informationOn the Ordinary and Signed Göllnitz-Gordon Partitions
On the Ordinary and Signed Göllnitz-Gordon Partitions Andrew V. Sills Department of Mathematical Sciences Georgia Southern University Statesboro, Georgia, USA asills@georgiasouthern.edu Version of October
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 informationRamanujan s theories of elliptic functions to alternative bases, and beyond.
Ramanuan s theories of elliptic functions to alternative bases, and beyond. Shaun Cooper Massey University, Auckland Askey 80 Conference. December 6, 03. Outline Sporadic sequences Background: classical
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 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 informationarxiv: v1 [math.nt] 28 Dec 2018
THE CONVERGENCE AND DIVERGENCE OF q-continued FRACTIONS OUTSIDE THE UNIT CIRCLE arxiv:82.29v [math.nt] 28 Dec 208 DOUGLAS BOWMAN AND JAMES MC LAUGHLIN Abstract. We consider two classes of q-continued fraction
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 information#A22 INTEGERS 17 (2017) NEW CONGRUENCES FOR `-REGULAR OVERPARTITIONS
#A22 INTEGERS 7 (207) NEW CONGRUENCES FOR `-REGULAR OVERPARTITIONS Shane Chern Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania shanechern@psu.edu Received: 0/6/6,
More informationON THE DIVERGENCE OF THE ROGERS-RAMANUJAN CONTINUED FRACTION ON THE UNIT CIRCLE
ON THE DIVERGENCE OF THE ROGERS-RAMANUJAN CONTINUED FRACTION ON THE UNIT CIRCLE D. BOWMAN AND J. MC LAUGHLIN Abstract. This paper is an intensive study of the convergence of the Rogers-Ramanujan continued
More informationFURTHER RESULTS FOR PARTITIONS INTO FOUR SQUARES OF EQUAL PARITY. Michael D. Hirschhorn and James A. Sellers
FURTHER RESULTS FOR PARTITIONS INTO FOUR SQUARES OF EQUAL PARITY Michael D. Hirschhorn James A. Sellers School of Mathematics UNSW Sydney 2052 Australia Department of Mathematics Penn State University
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 informationRamanujan-type congruences for broken 2-diamond partitions modulo 3
Progress of Projects Supported by NSFC. ARTICLES. SCIENCE CHINA Mathematics doi: 10.1007/s11425-014-4846-7 Ramanujan-type congruences for broken 2-diamond partitions modulo 3 CHEN William Y.C. 1, FAN Anna
More informationSrinivasa Ramanujan Life and Mathematics
Life and Mathematics Universität Wien (1887 1920) (1887 1920) (1887 1920) (1887 1920) (1887 1920) (1887 1920) (1887 1920) born 22 December 1887 in Erode (Madras Presidency = Tamil Nadu) named Iyengar (1887
More informationBruce C. Berndt and Ken Ono Dedicated to our good friend George Andrews on his 60th birthday. Introduction
RAMANUJAN S UNPUBLISHED MANUSCRIPT ON THE PARTITION AND TAU FUNCTIONS WITH PROOFS AND COMMENTARY Bruce C. Berndt and Ken Ono Dedicated to our good friend George Andrews on his 60th birthday Introduction
More informationarxiv:math/ v1 [math.ca] 8 Nov 2003
arxiv:math/0311126v1 [math.ca] 8 Nov 2003 PARTIAL SUMS OF HYPERGEOMETRIC SERIES OF UNIT ARGUMENT 1 WOLFGANG BÜHRING Abstract. The asymptotic behaviour of partial sums of generalized hypergeometric series
More information