EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 1-9 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Invited Paper Some Theta-Function Identities Related to Jacobi’s Triple-Product Identity H. M. Srivastava1,2,∗, M. P. Chaudhary3 and Sangeeta Chaudhary4 1 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada 2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China 3 Former Researcher, Department of Applied Sciences and Humanities, Jamia Millia Islamia, New Delhi 110025, India 4 School of Computational and Integrative Sciences, Jawaharlal Nehru University New Delhi 110067, India Abstract. The main object of this paper is to present some q-identities involving some of the theta functions of Jacobi and Ramanujan. These q-identities reveal certain relationships among three of the theta-type functions which arise from the celebrated Jacobi’s triple-product identity in a remarkably simple way. The results presented in this paper are motivated by some recent works by Chaudhary et al. (see [4] and [5]) and others (see, for example, [1] and [13]). 2010 Mathematics Subject Classifications: 11F27, 33E20 Key Words and Phrases: q-Identities, Infinite series and infinite products, Theta-function identities, Ramanujan’s general theta function, Jacobi’s triple-product identity 1. Introduction, Definitions and Preliminaries As long ago as 1829, Carl Gustav Jacob Jacobi (1804-1851) introduced a set of four theta functions ϑj(z, q) (j = 1, 2, 3, 4), which we recall here in the following forms (see [9] and [15, pp. 463 et seq.]; see also [12, p. 86]): ϑ1(z, q) = −i ∞∑ n=−∞ (−1)n e(2n+1)iz ∗Corresponding author. Email addresses: harimsri@math.uvic.ca (H. M. Srivastava), dr.m.p.chaudhary@gmail.com (M. P. Chaudhary), sangeeta.ch289@gmail.com Sangeeta Chaudhary http://www.ejpam.com 1 c© 2018 EJPAM All rights reserved. H. M. Srivastava, M. P. Chaudhary, S. Chaudhary / Eur. J. Pure Appl. Math, 11 (1) (2018), 1-9 2 = 2 ∞∑ n=0 (−1)n q ( n+1 2 2 )2 sin[(2n+ 1)z] = 2q 1 4 ∞∑ n=0 (−1)n qn(n+1) sin[(2n+ 1)z], (1) ϑ2(z, q) = ∞∑ n=−∞ q ( n+1 2 2 )2 e(2n+1)iz = 2 ∞∑ n=0 q ( n+1 2 2 )2 cos[(2n+ 1)z] = 2q 1 4 ∞∑ n=0 qn(n+1) cos[(2n+ 1)z], (2) ϑ3(z, q) = ∞∑ n=−∞ qn 2 e2niz = 1 + 2 ∞∑ n=1 qn 2 cos(2nz) (3) and ϑ4(z, q) = ∞∑ n=−∞ (−1)n qn 2 e2niz = 1 + 2 ∞∑ n=1 (−1)n qn 2 cos(2nz), (4) where z ∈ C and |q| < 1, C being the set of complex numbers. We also recall here that, in Chapter 16 of his celebrated Notebooks, Srinivasa Ramanujan (1887-1920) defined the general theta function f(a, b) as follows (see, for example, [10] and [11]; see also [1] and [13]): f(a, b) := 1 + ∞∑ n=1 (ab) n(n−1) 2 (an + bn) = ∞∑ n=−∞ a n(n+1) 2 b n(n−1) 2 = f(b, a) (|ab| < 1), (5) so that, for any integer n, it is easily seen that f(a, b) = a n(n+1) 2 b n(n−1) 2 f ( a(ab)n, b (ab)n ) = f(b, a). (6) Ramanujan also rediscovered Jacobi’s famous triple-product identity which, in Ramanu- jan’s notation, is given by f(a, b) = (−a; ab)∞ (−b; ab)∞ (ab; ab)∞, (7) H. M. Srivastava, M. P. Chaudhary, S. Chaudhary / Eur. J. Pure Appl. Math, 11 (1) (2018), 1-9 3 or, equivalently, by (see [9]) ∞∑ n=−∞ qn 2 zn = ∞∏ n=1 ( 1− q2n ) ( 1 + zq2n−1 )( 1 + 1 z q2n−1 ) = ( q2; q2 ) ∞ ( −zq; q2 ) ∞ ( −q z ; q2 ) ∞ (|q| < 1; z 6= 0), (8) which was, in fact, first proved by Carl Friedrich Gauss (1777-1855). As usual, in the above equations as well as throughout our present investigation, we denote the set of complex numbers by C and the set of positive integers by N with, of course, N0 := N∪{0}. Moreover, for q, λ, µ ∈ C (|q| < 1), the basic (or q-) shifted factorial (λ; q)µ is defined by (see, for example, [12, Chapter 3, Section 3.2.1] and [14, pp. 346 et seq.]) (λ; q)µ := ∞∏ j=0 ( 1− λqj 1− λqµ+j ) (|q| < 1; λ, µ ∈ C), (9) so that (λ; q)n :=  1 (n = 0) n−1∏ j=0 ( 1− λqj ) (n ∈ N) (10) and (λ; q)∞ := ∞∏ j=0 ( 1− λ qj ) (|q| < 1; λ ∈ C). (11) The theory of Jacobi’s theta functions ϑj(z, q) (j = 1, 2, 3, 4), which are defined above by the equations (1) to (4), has a long history and many applications in a wide variety of research fields such as number theory (especially in quadratic forms and elliptic functions) and quantum physics. Besides, the subject of q-analysis, which is popularly known as the quantum analysis, has its roots in such important areas as (for example) Mathematical Physics, Analytic Number Theory, and the Theory of Partitions. Motivated essentially by the potential for applications of q-series and q-products, we investigate here the following three most interesting functions which are related closely to such entities as Jacobi’s theta functions in the equations (1) to (4), Ramanujan’s general theta function in (5) and Jacobi’s triple-product identity in (7) or (8) (see also [1] and [13]): f(−q) := f(−q,−q2) = ∞∑ n=−∞ (−1)n q n(3n−1) 2 = (q; q)∞ = 1√ 3 q− 1 24 ϑ2 (π 6 , x 1 6 ) , (12) H. M. Srivastava, M. P. Chaudhary, S. Chaudhary / Eur. J. Pure Appl. Math, 11 (1) (2018), 1-9 4 ϕ(q) := f(q, q) = ∞∑ n=−∞ qn 2 = (−q; q2)∞ (q2; q2)∞ (q; q2)∞ (−q2; q2)∞ = ϑ3(0, q) (13) and ψ(q) := f(q, q3) = ∞∑ n=0 q n(n+1) 2 = (q2; q2)∞ (q; q2)∞ = 1 2 q− 1 8 [ ϑ2 ( 0, √ q ) − 1 ] . (14) The main object of the present article is to prove two (presumably new) q-identities which provide interesting relationships among the above-defined three ϑ-type functions f(−q), ϕ(q) and ψ(q), each of which arises from Jacobi’s triple-product identity (8) in a remarkably simple way. For more details and further results, the interested reader may be referred to the works presented in [1], [2], [3], [6], [7], [8] and [13]. 2. The Main Results In this section, we begin by expressing the functions f(−q), ϕ(q) and ψ(q) in rising powers of q as follows: f(−q) = 1 + ∞∑ n=1 (−1)n ( q n(3n−1) 2 + q n(3n+1) 2 ) = 1− q − q2 + q5 + q7 − q12 − q15 + q22 + q26 − · · · , (15) ϕ(q) = 1 + 2 ∞∑ n=1 qn 2 = 1 + 2q + 2q4 + 2q9 + 2q16 + · · · (16) and ψ(q) = 1 + ∞∑ n=1 q n(n+1) 2 = 1 + q + q3 + q6 + q10 + q15 + · · · . (17) We now state our main results as the following Theorem. Theorem. Each of the following relationships holds true: 2qf(−q3)ψ ( q9 ) = f ( −q6 ) [ ϕ ( −q9 ) − ϕ(−q) ] (18) and 2f(−q)f ( −q2 ) = ϕ ( q4 ) [ ψ(q) + qψ ( q9 )] + ϕ ( q36 ) [ ψ(q)− 3qψ ( q9 )] H. M. Srivastava, M. P. Chaudhary, S. Chaudhary / Eur. J. Pure Appl. Math, 11 (1) (2018), 1-9 5 − 2qψ(q) [ ψ ( q8 ) + q8ψ ( q72 )] − 2q2ψ ( q9 ) [ ψ ( q8 ) − 3q8ψ ( q72 )] , (19) where the functions f(q), ϕ(q) and ψ(q) are given by (12), (13) and (14), respectively. Proof. First of all, we shall prove our first q-identity (18). Let L1(q) and R1(q) denote the left-hand and the right-hand sides of the q-identity (18), respectively. Then, in order to compute the value for L1(q), by using (15) (for q 7→ q3) and (17) (for q 7→ q9), we have L1(q) = 2q ( 1− q3 − q6 + q15 + q21 − · · · ) · ( 1 + q9 + q27 + q54 + q90 + q135 + q189 + q252 + · · · ) , which, after multiplication and further simplification, yields L1(q) = 2q − 2q4 − 2q7 + 2q10 − 2q13 + 2q22 + 2q25 + 2q28 − 2q34 − 2q37 + 2q43 − 4q46 + 2q49 − 2q58 − 2q61 − 2q64 + 2q67 + 2q70 − 2q73 + 4q76 + 2q79 + 2q88 − 2q97 − 2q100 + · · · . (20) In a similar way, we can compute the value for R1(q) by applying (15) (for q 7→ q6) and (16) (for q 7→ −q and q 7→ −q9) as follows: R1(q) = ( 1− q6 − q12 + q30 + q42 − q72 − q90 + · · · ) · [ ( 1− 2q9 + 2q36 − 2q81 + 2q144 − 2q225 + 2q324 − 2q441 + 2q576 − · · · ) − ( 1− 2q + 2q4 − 2q9 + 2q16 − 2q25 + 2q36 − 2q49 + 2q64 − 2q81 + · · · ) ] , which, after simplification and by using algebraic manipulation, becomes R1(q) = 2q − 2q4 − 2q7 + 2q10 − 2q13 + 2q22 + 2q25 + 2q28 − 2q34 − 2q37 + 2q43 − 4q46 + 2q49 − 2q58 − 2q61 − 2q64 + 2q67 + 2q70 − 2q73 + 4q76 + 2q79 + 2q88 − 2q97 − 2q100 + · · · . (21) By comparing the equations (20) and (21), we readily arrive at the q-identity (18). We next prove the second q-identity (19). Let L2(q) and R2(q) denote the left-hand and the right-hand sides of (19), respectively. Then, in order to compute the value for L2(q), we make use of (15) (for q 7→ q and q 7→ q2) as follows: L2(q) = 2 ( 1− q − q2 + q5 + q7 − q12 − q15 + · · · ) · ( 1− q2 − q4 + q10 + q14 − q24 − q30 + q44 + q52 − · · · ) , which, after simplification and by using algebraic manipulation, yields L2(q) = 2 ( 1− q − 2q2 + q3 + 2q5 + q6 − 2q9 + q10 − 2q11 − 2q12 + 2q14 − q15 + 2q17 + 2q19 + q21 − 2q24 − q28 − 2q29 − 2q30 + 2q32 − 2q35 + 3q36 + 2q39 + 2q42 + 2q44 − q45 − 2q46 − 2q50 + 2q51 − 2q53 − 2q54 − q55 − 2q56 H. M. Srivastava, M. P. Chaudhary, S. Chaudhary / Eur. J. Pure Appl. Math, 11 (1) (2018), 1-9 6 + 2q57 + q59 − q60 + 2q65 + q66 + 2q71 + 2q72 + 2q74 − · · · ) . (22) We now compute the value for ϕ ( q4 ) [ ψ(q) + qψ ( q9 )] , which occurs in (19). By applying (16) (for q 7→ q4) and (17) (for q 7→ q and q 7→ q9), we have ϕ ( q4 ) [ ψ(q) + qψ ( q9 )] = (1 + 2q4 + 2q16 + 2q36 + 2q64 + · · · ) · [( 1 + q + q3 + q6 + q10 + q15 + · · · ) + q ( 1 + q9 + q27 + q54 + q90 + q135 + · · · )] , which, after simplification and by using algebraic manipulation, assumes the following form: ϕ ( q4 ) [ ψ(q) + qψ ( q9 )] = 1 + 2q + q3 + 2q4 + 4q5 + q6 + 2q7 + 4q10 + 4q14 + q15 + 2q16 + 4q17 + 4q19 + q21 + 2q22 + 2q25 + 4q26 + 2q28 + 2q31 + 4q32 + 3q36 + 6q37 + 2q39 + 2q40 + 2q42 + 4q44 + q45 + 4q46 + 2q49 + 2q51 + 2q52 + 2q55 + · · · . (23) We next compute the value for ϕ ( q36 ) [ ψ(q)− 3qψ ( q9 )] , which occurs in (19). By applying (16) (for q 7→ q36) and (17) (for q 7→ q and q 7→ q9), we find that ϕ ( q36 ) [ ψ(q)− 3qψ ( q9 )] = ( 1 + 2q36 + 2q144 + 2q324 + · · · ) · [ ( 1 + q + q3 + q6 + q10 + q15 + · · · ) − 3q ( 1 + q9 + q27 + q54 + q90 + q135 + · · · ) ] , which, after simplification and by using algebraic manipulation, yields ϕ ( q36 ) [ ψ(q)− 3qψ ( q9 )] = 1− 2q + q3 + q6 − 2q10 + q15 + q21 − 2q28 + 3q36 − 4q37 + 2q39 + 2q42 + q45 − 4q46 + 2q51 − 2q55 + 2q57 − 4q64 + q66 + 2q72 + · · · . (24) In order to compute the value for −2qψ(q) [ ψ ( q8 ) + q8ψ ( q72 )] , which occurs in (19), by making use of (17) (for q 7→ q, q 7→ q8 and q 7→ q72), we obtain − 2qψ(q) [ ψ ( q8 ) + q8ψ ( q72 )] = −2q ( 1 + q + q3 + q6 + q10 + q15 + · · · ) H. M. Srivastava, M. P. Chaudhary, S. Chaudhary / Eur. J. Pure Appl. Math, 11 (1) (2018), 1-9 7 · [( 1 + q8 + q24 + q48 + q80 + q120 + · · · ) + q8 ( 1 + q72 + q216 + q432 + · · · )] , which, after simplification and by using algebraic manipulation, becomes − 2qψ(q) [ ψ ( q8 ) + q8ψ ( q72 )] = −2q − 2q2 − 2q4 − 2q7 − 4q9 − 4q10 − 2q11 − 4q12 − 4q15 − 2q16 − 4q19 − 2q22 − 4q24 − 2q25 − 2q26 − 2q28 − 2q29 − 4q30 − 2q31 − 2q35 − 6q37 − 2q40 − 4q45 − 4q46 − 2q49 − 2q50 − 2q52 − 2q53 − 4q54 − 2q55 − 2q56 − 2q59 − 2q61 − 6q64 − · · · . (25) Finally, we compute the value for −2q2ψ ( q9 ) [ ψ ( q8 ) − 3q8ψ ( q72 )] , which occurs in (19). By making use of (17) (for q 7→ q8, q 7→ q9 and q 7→ q72), it is easily observed that − 2q2ψ ( q9 ) [ ψ ( q8 ) − 3q8ψ ( q72 )] = −2q2 ( 1 + q9 + q27 + q54 + q90 + q135 + q189 + · · · )[( 1 + q8 + q24 + q48 + q80 + q120 + · · · ) − 3q8(1 + q72 + q216 + q432 + · · · ) ] , which, after simplification and by using algebraic manipulation, yields − 2q2ψ ( q9 ) [ ψ ( q8 ) − 3q8ψ ( q72 )] = −2q2 + 4q10 − 2q11 + 4q19 − 2q26 − 2q29 − 2q35 + 4q37 − 2q50 − 2q53 − 2q56 − 2q59 + 4q64 − 2q77 − 2q80 + · · · . (26) Thus, by applying the equations (23) to (26), we find that R2(q) = ϕ ( q4 ) [ ψ(q) + qψ ( q9 )] + ϕ ( q36 ) [ ψ(q)− 3qψ ( q9 )] − 2qψ(q) [ ψ ( q8 ) + q8ψ ( q72 )] − 2q2ψ ( q9 ) [ ψ ( q8 ) − 3q8ψ ( q72 )] = 2 [ 1− q − 2q2 + q3 + 2q5 + q6 − 2q9 + q10 − 2q11 − 2q12 + 2q14 − q15 + 2q17 + 2q19 + q21 − 2q24 − q28 − 2q29 − 2q30 + 2q32 − 2q35 + 3q36 + 2q39 + 2q42 + 2q44 − q45 − 2q46 − 2q50 + 2q51 − 2q53 − 2q54 − q55 − 2q56 + 2q57 + q59 − q60 + 2q65 + q66 + 2q71 + 2q72 + 2q74 − · · · ] . (27) The q-identity (19) now follows upon comparing the equations (22) and (27). We thus have completed our proof of the Theorem. 3. Concluding Remarks and Observations Our present article is motivated essentially by the potential for applications of q-series and q-products. We have investigated here the three most interesting functions f(q), ϕ(q) REFERENCES 8 and ψ(q), which are related closely to such celebrated entities as Jacobi’s theta func- tions in the equations (1) to (4), Ramanujan’s general theta function in (5) and Jacobi’s triple-product identity in (7) or (8). Our main results are stated and proved as the above Theorem and provide a sequel to some recent works by Chaudhary et al. (see [4] and [5]), by Adiga et al. [1], and by Srivastava and Chaudhary [13]. Acknowledgements The third-named author (Sangeeta Chaudhary) is thankful to the National Board of Higher Mathematics (NBHM) under the Department of Atomic Energy (DAE) of the Government of India for providing financial support by awarding her a Post-Doctoral Fellowship (Grant Numbers: 2/40(47)/2015/R and D-II/8417 dated 22 June 2017) while carrying out this research work. References [1] C. Adiga, N. A. S. Bulkhali, D. Ranganatha and H. M. Srivastava, Some new modular relations for the Rogers-Ramanujan type functions of order eleven with applications to partitions, J. Number Theory 158 (2016), 281–297. [2] M. P. Chaudhary, On q-product identities, Pacific J. Appl. Math. 5 (2013), 123–129. [3] M. P. Chaudhary and J. Choi, Note on modular relations for Rogers-Ramanujan type identities and representation for Jacobi identities, East Asian Math. J. 31 (2015), 659–665. [4] M. P. Chaudhary, G. A. Salilew and J. Choi, Two theta function identities, Far East J. Math. Sci. 101 (2017), 1833–1837. 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