Mathematical Analysis and Analytic Number Theory 2019 Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Rekha Srivastava Edited by Mathematical Analysis and Analytic Number Theory 2019 Mathematical Analysis and Analytic Number Theory 2019 Editor Rekha Srivastava MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Rekha Srivastava Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4 Canada Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Mathematical Analysis Analytic Number Theory 2019). 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Contents About the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to ”Mathematical Analysis and Analytic Number Theory 2019” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Hari Mohan Srivastava, Rekha Srivastava, Mahendra PalChaudhary and Salah Uddin A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity Reprinted from: Mathematics 2020 , 8 , 918, doi:10.3390/math8060918 . . . . . . . . . . . . . . . . . 1 Muhammad Naeem, Saqib Hussain, Shahid Khan, Tahir Mahmood, Maslina Darus and Zahid Shareef Janowski Type q -Convex and q - Close-to-Convex Functions Associated with q -Conic Domain Reprinted from: Mathematics 2020 , 8 , 440, doi:10.3390/math8030440 . . . . . . . . . . . . . . . . . 15 Davood Alimohammadi, Nak Eun Cho, Ebrahim Analouei Adegani and Ahmad Motamednezhad Argument and Coefficient Estimates for Certain Analytic Functions Reprinted from: Mathematics 2020 , 8 , 88, doi:10.3390/math8010088 . . . . . . . . . . . . . . . . . . 29 Mohd Qasim, M. Mursaleen, Asif Khan and Zaheer Abbas Approximation by Generalized Lupas ̧ Operators Based on q -Integers Reprinted from: Mathematics 2020 , 8 , 68, doi:10.3390/math8010068 . . . . . . . . . . . . . . . . . . 43 Qaiser Khan, Muhammad Arif, Mohsan Raza, Gautam Srivastava, Huo Tang and Shafiq ur Rehman Some Applications of a New Integral Operator in q -Analog for Multivalent Functions Reprinted from: Mathematics 2019 , 7 , 1178, doi:10.3390/math7121178 . . . . . . . . . . . . . . . . 59 Hari Mohan Srivastava, Asifa Tassaddiq, Gauhar Rahman, Kottakkaran Sooppy Nisar and Ilyas Khan A New Extension of the τ -GaussHypergeometric Function and ItsAssociated Properties Reprinted from: Mathematics 2019 , 7 , 996, doi:10.3390/math7100996 . . . . . . . . . . . . . . . . . 73 Nak Eun Cho, Ebrahim Analouei Adegani, Serap Bulut and Ahmad Motamednezhad The Second Hankel Determinant Problem for a Class of Bi-Close-to-ConvexFunctions Reprinted from: Mathematics 2019 , 7 , 986, doi:10.3390/math7100986 . . . . . . . . . . . . . . . . . 83 Zhaolin Jiang, Weiping Wang, Yanpeng Zheng, Baishuai Zuo and Bei Niu Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices Reprinted from: Mathematics 2019 , 7 , 939, doi:10.3390/math7100939 . . . . . . . . . . . . . . . . . 93 Yunlan Wei, Yanpeng Zheng, Zhaolin Jiang and Sugoog Shon A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers Reprinted from: Mathematics 2019 , 7 , 893, doi:10.3390/math7100893 . . . . . . . . . . . . . . . . . 111 v Hari M. Srivastava, Qazi Zahoor Ahmad, Maslina Darus, Nazar Khan, Bilal Khan, Naveed Zaman and Hasrat Hussain Shah Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated withthe Lemniscate of Bernoulli Reprinted from: Mathematics 2019 , 7 , 848, doi:10.3390/math7090848 . . . . . . . . . . . . . . . . . 123 Namhoon Kim Transformation of Some Lambert Series and Cotangent Sums Reprinted from: Mathematics 2019 , 7 , 840, doi:10.3390/math7090840 . . . . . . . . . . . . . . . . . 133 Anthony Sofo and Amrik Singh Nimbran Euler Sums and Integral Connections Reprinted from: Mathematics 2019 , 7 , 833, doi:10.3390/math7090833 . . . . . . . . . . . . . . . . . 143 Pierpaolo Natalini and Paolo Emilio Ricci Appell-Type Functions and Chebyshev Polynomials Reprinted from: Mathematics 2019 , 7 , 679, doi:10.3390/math7080679 . . . . . . . . . . . . . . . . . 167 Lei Shi, Qaiser Khan, Gautam Srivastava, Jin-Lin Liu, and Muhammad Arif A Study of Multivalent q -starlike Functions Connected with Circular Domain Reprinted from: Mathematics 2019 , 7 , 670, doi:10.3390/math7080670 . . . . . . . . . . . . . . . . . 175 Konstantinos Kalimeris and Athanassios S. Fokas A Novel Integral Equation for the Riemann Zeta Function and Large t -Asymptotics Reprinted from: Mathematics 2019 , 7 , 650, doi:10.3390/math7070650 . . . . . . . . . . . . . . . . . 187 Aoen and Shuhai Li The Application of Generalized Quasi-Hadamard Products of Certain Subclasses of Analytic Functions with Negative and Missing Coefficients Reprinted from: Mathematics 2019 , 7 , 620, doi:10.3390/math7070620 . . . . . . . . . . . . . . . . . 205 Giuseppe Dattoli, Silvia Licciardi, Elio Sabia, Hari M. Srivastava Some Properties and Generating Functions ofGeneralized Harmonic Numbers Reprinted from: Mathematics 2019 , 7 , 577, doi:10.3390/math7070577 . . . . . . . . . . . . . . . . . 215 Lina Ma, Shuhai Li and Xiaomeng Niu Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination Reprinted from: Mathematics 2019 , 7 , 548, doi:10.3390/math7060548 . . . . . . . . . . . . . . . . . 227 Lei Shi, Izaz Ali,Muhammad Arif, Nak Eun Cho, Shehzad Hussain, and HassanKhan A Study of Third Hankel Determinant Problem for Certain Subfamilies of Analytic Functions Involving Cardioid Domain Reprinted from: Mathematics 2019 , 7 , 418, doi:10.3390/math7050418 . . . . . . . . . . . . . . . . . 245 Chenkuan Li, Changpin Li, Thomas Humphries and Hunter Plowman Remarks on the Generalized Fractional Laplacian Operator Reprinted from: Mathematics 2019 , 7 , 320, doi:10.3390/math7040320 . . . . . . . . . . . . . . . . . 261 Khurshid Ahmad, Saima Mustafa, Muhey U. Din, Shafiq Ur Rehman, Mohsan Raza and Muhammad Arif On Geometric Properties of NormalizedHyper-Bessel Functions Reprinted from: Mathematics 2019 , 7 , 316, doi:10.3390/math7040316 . . . . . . . . . . . . . . . . . 279 vi Shahid Mahmood, Janusz Sok ́ oł, Hari Mohan Srivastav and Sarfraz Nawaz Malik Some Reciprocal Classes of Close-to-Convex and Quasi-Convex Analytic Functions Reprinted from: Mathematics 2019 , 7 , 309, doi:10.3390/math7040309 . . . . . . . . . . . . . . . . . 291 vii About the Editor Rekha Srivastava was born in the small town of Chikati (District Ganjam) in the Province of Odisha in India. After completing her high school and intermediate college education, she received her B.Sc. degree in 1962 from Utkal University in the Province of Odisha. She then travelled to Banaras Hindu University in Varanasi in the Province of Uttar Pradesh in India, from where she first received her M.Sc. degree in 1965 and then her Ph.D. degree in 1967 at the age of 22. In fact, she happens to be the first woman in the entire Province of Odisha to have received a Ph.D. degree in Mathematics. She began her university-level teaching career in 1968 and, after having taught in India until 1972, she joined the University of Victoria in Canada, where she held the positions of Postdoctoral Research Fellow, Research Associate, and Visiting Scientist until the year 1977. Currently, she is a Retired Professor (Adjunct), having taught at the University of Victoria since 1977 until her retirement. In the year 1997, Professor Srivastava was nominated for the Best Teacher Award in the Faculty of Science at the University of Victoria. Professor Rekha Srivastava is currently associated with many scientific research journals as Editor or Guest Editor. She has published about 90 original papers in peer-reviewed international scientific research journals. Her current Google Scholar h-index is 24. She received the Distinguished Service Award, which was formally awarded to her on 14 December 2015 at the Eighteenth Annual Conference and the First International Conference of the Vijnana Parishad (Science Academy) of India at the Maulana Azad National Institute of Technology in Bhopal in the Province of Madhya Pradesh in India, which was held on 11–14 December 2015, for her outstanding contributions to Mathematics and for her distinguished services rendered to the Vijnana Parishad (Science Academy) of India. She also received the Fellowship Award (F.V.P.I.) on 26 November 2017 at the Twentieth Annual Conference of the Vijnana Parishad (Science Academy) of India at Manipal University in Jaipur in the Province of Rajasthan in India, which was held on 24–26 November 2017, for her outstanding professional contributions and scholarly achievements in Mathematics and Its Applications. Moreover, as a Member of the Organizing and Scientific Committees and also as an Invited Speaker, Professor Srivastava has participated in and delivered many lectures on her research work at a large number of international conferences held around the world. Professor Rekha Srivastava’s research interests include several areas of pure and applied mathematical sciences, such as (for example) real and complex analysis, analytic number theory, fractional calculus and its applications, integral equations and integral transforms, and special functions and their applications. Further details about Professor Rekha Srivastava’s professional achievements and scholarly accomplishments, as well as honors, awards, and distinctions, can be found at the following website: http://www.math.uvic.ca/ ∼ rekhas/. ix Preface to ”Mathematical Analysis and Analytic Number Theory 2019” This volume contains a total of 22 peer-reviewed and accepted submissions (including several invited feature articles) from all over the world to the Special Issue of the MDPI journal, Mathematics , on the general subject area of “Mathematical Analysis and Analytic Number Theory”. This volume contains a total of 22 peer-reviewed and accepted submissions (including several invited feature articles) from all over the world to the Special Issue of the MDPI journal, Mathematics, on the general subject area of “Mathematical Analysis and Analytic Number Theory”. The suggested topics of interest for the call of papers for this Special Issue included, but were not limited to, the following themes: • Theory and applications of the tools and techniques of mathematical analysis • Theory and applications of the tools and techniques of analytic number theory • Applications involving special (or higher transcendental) functions • Applications involving fractional-order differential and differintegral equations • Applications involving q-series and q-polynomials • Applications involving special functions of mathematical physics and applied mathematics • Applications involving geometric function theory of complex analysis • Applications involving real analysis and operator theory Finally, it gives me great pleasure to thank all of the participating authors in this Special Issue as well as the editorial personnel of the MDPI Editorial Office for Mathematics for their invaluable input and contributions toward the success of this Special Issue. The wholehearted support and dedication of one and all are indeed greatly appreciated. Rekha Srivastava Editor xi mathematics Article A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity Hari Mohan Srivastava 1,2,3 , Rekha Srivastava 1, *, Mahendra Pal Chaudhary 4 and Salah Uddin 5 1 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada; harimsri@math.uvic.ca 2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 3 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan 4 Department of Mathematics, Netaji Subhas University of Technology, Sector 3, Dwarka, New Delhi 110078, India; dr.m.p.chaudhary@gmail.com 5 Department of Mathematics, PDM University, Bahadurgarh 124507, Haryana State, India; vsludn@gmail.com * Correspondence: rekhas@math.uvic.ca Received: 11 May 2020; Accepted: 3 June 2020; Published: 5 June 2020 Abstract: The authors establish a set of six new theta-function identities involving multivariable R -functions which are based upon a number of q -product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper, we consider and relate the multivariable R -functions to several interesting q -identities such as (for example) a number of q -product identities and Jacobi’s celebrated triple-product identity. Various recent developments on the subject-matter of this article as well as some of its potential application areas are also briefly indicated. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities and present a presumably open problem. Keywords: theta-function identities; multivariable R -functions; Jacobi’s triple-product identity; Ramanujan’s theta functions; q -product identities; Euler’s pentagonal number theorem; Rogers-Ramanujan continued fraction; Rogers-Ramanujan identities; combinatorial partition-theoretic identities; Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities; Schur’s second partition theorem 1. Introduction and Definitions Throughout this article, we denote by N , Z , and C the set of positive integers, the set of integers and the set of complex numbers, respectively. We also let N 0 : = N ∪ { 0 } = { 0, 1, 2, · · · } In what follows, we shall make use of the following q -notations for the details of which we refer the reader to a recent monograph on q -calculus by Ernst [ 1 ] and also to the earlier works on the subject by Slater [ 2 ] (Chapter 3, Section 3.2.1), and by Srivastava et al. ( [ 3 ] (pp. 346 et seq.) and [ 4 ] (Chapter 6) ) Mathematics 2020 , 8 , 918; doi:10.3390/math8060918 www.mdpi.com/journal/mathematics 1 Mathematics 2020 , 8 , 918 The q -shifted factorial ( a ; q ) n is defined (for | q | < 1) by ( a ; q ) n : = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 ( n = 0 ) n − 1 ∏ k = 0 ( 1 − aq k ) ( n ∈ N ) , ( 1 ) where a , q ∈ C , and it is assumed tacitly that a = q − m ( m ∈ N 0 ) . We also write ( a ; q ) ∞ : = ∞ ∏ k = 0 ( 1 − aq k ) = ∞ ∏ k = 1 ( 1 − aq k − 1 ) ( a , q ∈ C ; | q | < 1 ) ( 2 ) It should be noted that, when a = 0 and | q | 1, the infinite product in Equation (2) diverges. Thus, whenever ( a ; q ) ∞ is involved in a given formula, the constraint | q | < 1 will be tacitly assumed to be satisfied. The following notations are also frequently used in our investigation: ( a 1 , a 2 , · · · , a m ; q ) n : = ( a 1 ; q ) n ( a 2 ; q ) n · · · ( a m ; q ) n ( 3 ) and ( a 1 , a 2 , · · · , a m ; q ) ∞ : = ( a 1 ; q ) ∞ ( a 2 ; q ) ∞ · · · ( a m ; q ) ∞ ( 4 ) Ramanujan (see [ 5 , 6 ]) defined the general theta function f ( a , b ) as follows (see, for details, in [7] (p. 31, Equation (18.1)) and [8,9]): f ( a , b ) = 1 + ∞ ∑ n = 1 ( ab ) n ( n − 1 ) 2 ( a n + b n ) = ∞ ∑ n = − ∞ a n ( n + 1 ) 2 b n ( n − 1 ) 2 = f ( b , a ) ( | ab | < 1 ) (5) We find from this last Equation (5) that f ( a , b ) = a n ( n + 1 ) 2 b n ( n − 1 ) 2 f ( a ( ab ) n , b ( ab ) − n ) = f ( b , a ) ( n ∈ Z ) ( 6 ) In fact, Ramanujan (see [ 5 , 6 ]) also rediscovered Jacobi’s famous triple-product identity, which, in Ramanujan’s notation, is given by (see [7] (p. 35, Entry 19)): f ( a , b ) = ( − a ; ab ) ∞ ( − b ; ab ) ∞ ( ab ; ab ) ∞ ( 7 ) or, equivalently, by (see [10]) ∞ ∑ n = − ∞ q n 2 z n = ∞ ∏ n = 1 ( 1 − q 2 n ) ( 1 + zq 2 n − 1 ) ( 1 + 1 z q 2 n − 1 ) = ( q 2 ; q 2 ) ∞ ( − zq ; q 2 ) ∞ ( − q z ; q 2 ) ∞ ( | q | < 1; z = 0 ) Remark 1. Equation ( 6 ) holds true as stated only if n is any integer. In case n is not an integer, this result (6) is only approximately true ( see, for details, [ 5 ] (Vol. 2, Chapter XVI, p. 193, Entry 18 (iv)) ) Moreover, historically speaking , the q -series identity ( 7 ) or its above-mentioned equivalent form was first proved by Carl Friedrich Gauss (1777–1855). 2 Mathematics 2020 , 8 , 918 Several q -series identities, which emerge naturally from Jacobi’s triple-product identity (7), are worthy of note here (see, for details, (pp. 36–37, Entry 22) in [7]): φ ( q ) : = ∞ ∑ n = − ∞ q n 2 = 1 + 2 ∞ ∑ n = 1 q n 2 = { ( − q ; q 2 ) ∞ } 2 ( q 2 ; q 2 ) ∞ = ( − q ; q 2 ) ∞ ( q 2 ; q 2 ) ∞ ( q ; q 2 ) ∞ ( − q 2 ; q 2 ) ∞ ; (8) ψ ( q ) : = f ( q , q 3 ) = ∞ ∑ n = 0 q n ( n + 1 ) 2 = ( q 2 ; q 2 ) ∞ ( q ; q 2 ) ∞ ; ( 9 ) f ( − q ) : = f ( − q , − q 2 ) = ∞ ∑ n = − ∞ ( − 1 ) n q n ( 3 n − 1 ) 2 = ∞ ∑ n = 0 ( − 1 ) n q n ( 3 n − 1 ) 2 + ∞ ∑ n = 1 ( − 1 ) n q n ( 3 n + 1 ) 2 = ( q ; q ) ∞ (10) Equation (10) is known as Euler’s Pentagonal Number Theorem Remarkably, the following q -series identity: ( − q ; q ) ∞ = 1 ( q ; q 2 ) ∞ = 1 χ ( − q ) ( 11 ) provides the analytic equivalent form of Euler’s famous theorem (see, for details, [11,12]). Theorem 1. (Euler’s Pentagonal Number Theorem) The number of partitions of a given positive integer n into distinct parts is equal to the number of partitions of n into odd parts. We also recall the Rogers-Ramanujan continued fraction R ( q ) given by R ( q ) : = q 1 5 H ( q ) G ( q ) = q 1 5 f ( − q , − q 4 ) f ( − q 2 , − q 3 ) = q 1 5 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = q 1 5 1 + q 1 + q 2 1 + q 3 1 + ( | q | < 1 ) (12) Here, G ( q ) and H ( q ) , which are associated with the widely-investigated Roger-Ramanujan identities, are defined as follows: G ( q ) : = ∞ ∑ n = 0 q n 2 ( q ; q ) n = f ( − q 5 ) f ( − q , − q 4 ) = 1 ( q ; q 5 ) ∞ ( q 4; q 5 ) ∞ = ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ ( q 5 ; q 5 ) ∞ ( q ; q ) ∞ (13) and H ( q ) : = ∞ ∑ n = 0 q n ( n + 1 ) ( q ; q ) n = f ( − q 5 ) f ( − q 2 , − q 3 ) = 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ ( q 5 ; q 5 ) ∞ ( q ; q ) ∞ , (14) and the functions f ( a , b ) and f ( − q ) are given by Equations (5) and (10), respectively. 3 Mathematics 2020 , 8 , 918 For a detailed historical account of (and for various related developments stemming from) the Rogers-Ramanujan continued fraction (12) as well as the Rogers-Ramanujan identities (13) and (14), the interested reader may refer to the monumental work [7] (p. 77 et seq.) (see also [4,8]). The following continued-fraction results may be recalled now (see, for example, (p. 5, Equation (2.8)) in [13]). Theorem 2. Suppose that | q | < 1 . Then, A ( q ) : = ( q 2 ; q 2 ) ∞ ( − q ; q ) ∞ = ( q 2 ; q 2 ) ∞ ( q ; q 2 ) ∞ = 1 1 − q 1 + q ( 1 − q ) 1 − q 3 1 + q 2 ( 1 − q 2 ) 1 − q 5 1 + q 3 ( 1 − q 3 ) 1 − · · · = 1 1 − q 1 + q ( 1 − q ) 1 − q 3 1 + q 2 ( 1 − q 2 ) 1 − q 5 1 + q 3 ( 1 − q 3 ) 1 − · · · , (15) B ( q ) : = ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = 1 1 + q 1 + q 2 1 + q 3 1 + q 4 1 + q 5 1 + q 6 1 + · · · = 1 1 + q 1 + q 2 1 + q 3 1 + q 4 1 + q 5 1 + q 6 1 + · · · (16) and C ( q ) : = ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ = 1 + q 1 + q 2 1 + q 3 1 + q 4 1 + q 5 1 + q 6 1 + · · · = 1 + q 1 + q 2 1 + q 3 1 + q 4 1 + q 5 1 + q 6 1 + · · · (17) By introducing the general family R ( s , t , l , u , v , w ) , Andrews et al. [ 14 ] investigated a number of interesting double-summation hypergeometric q -series representations for several families of partitions and further explored the rôle of double series in combinatorial-partition identities: R ( s , t , l , u , v , w ) : = ∞ ∑ n = 0 q s ( n 2 )+ tn r ( l , u , v , w ; n ) , ( 18 ) 4 Mathematics 2020 , 8 , 918 where r ( l , u , v , w : n ) : = [ n u ] ∑ j = 0 ( − 1 ) j q uv ( j 2 )+( w − ul ) j ( q ; q ) n − uj ( q uv ; q uv ) j ( 19 ) We also recall the following interesting special cases of (18) (see, for details, (p. 106, Theorem 3) in [14]; see also [8]): R ( 2, 1, 1, 1, 2, 2 ) = ( − q ; q 2 ) ∞ , ( 20 ) R ( 2, 2, 1, 1, 2, 2 ) = ( − q 2 ; q 2 ) ∞ ( 21 ) and R ( m , m , 1, 1, 1, 2 ) = ( q 2 m ; q 2 m ) ∞ ( q m ; q 2 m ) ∞ ( 22 ) For the sake of brevity in our presentation of the main results, we now introduce the following notations: R α = R ( 2, 1, 1, 1, 2, 2 ) , R β = R ( 2, 2, 1, 1, 2, 2 ) and R m = R ( m , m , 1, 1, 1, 2 ) ( m ∈ N ) Ever since the year 2015, several new advancements and generalizations of the existing results were made in regard to combinatorial partition-theoretic identities (see, for example, [ 8 , 15 – 24 ]). In particular, Chaudhary et al. generalized several known results on character formulas (see [ 22 ]), Roger-Ramanujan type identities (see [ 19 ]), Eisenstein series, the Ramanujan-Göllnitz-Gordon continued fraction (see [ 20 ]), the 3-dissection property (see [ 18 ]), Ramanujan’s modular equations of degrees 3, 7, and 9 (see [ 16 , 17 ]), and so on, by using combinatorial partition-theoretic identities. An interesting recent investigation on the subject of combinatorial partition-theoretic identities by Hahn et al. [25] is also worth mentioning in this connection. Here, in this paper, our main objective is to establish a set of six new theta-function identities which depict the inter-relationships that exist between the multivariable R -functions, q -product identities, and partition-theoretic identities. Each of the following preliminary results will be needed for the demonstration of our main results in this paper (see [26] (pp. 1749–1750 and 1752–1754)): I. If P = ψ ( q ) q 1 2 ψ ( q 5 ) and Q = ψ ( q 3 ) q 3 2 ψ ( q 15 ) , then PQ + 5 PQ = ( Q P ) 2 − ( P Q ) 2 + 3 ( Q P + P Q ) ( 23 ) II. If P = ψ ( q ) q 1 4 ψ ( q 3 ) and Q = ψ ( q 5 ) q 5 4 ψ ( q 15 ) , then ( PQ ) 2 + ( 3 PQ ) 2 = ( Q P ) 3 − ( P Q ) 3 − 5 ( Q P − P Q ) + 5 ( P Q ) 2 + 5 ( Q P ) 2 (24) 5 Mathematics 2020 , 8 , 918 III. If P = ψ ( q ) q 1 4 ψ ( q 3 ) and Q = ψ ( q 7 ) q 7 4 ψ ( q 21 ) , then ( PQ ) 3 [( P Q ) 8 − 1 ] + 14 P 5 Q [( P Q ) 4 − 1 ] = P 6 Q 2 ( 7 − P 4 ) + 7 P 6 Q 2 ( P 4 − 3 ) − { 27 ( P Q ) 4 − 7 P 4 [ 3 + 3 ( P Q ) 4 − P 4 ]} (25) IV. If P = ψ ( q ) q 1 4 ψ ( q 3 ) and Q = ψ ( q 2 ) q 1 2 ; ψ ( q 6 ) , then ( P Q ) 2 + 3 P 2 − P 2 + ( Q P ) 2 = 0. ( 26 ) V. If P = ψ ( − q ) q 1 4 ψ ( − q 3 ) and Q = ψ ( q 2 ) q 1 2 ψ ( q 6 ) , then ( P Q ) 2 + 3 P 2 + P 2 − ( Q P ) 2 = 0. ( 27 ) VI. If P = ψ ( − q ) q 1 4 ψ ( − q 3 ) and Q = ψ ( q ) q 1 4 ψ ( q 3 ) , then [( P Q ) 2 − ( Q P ) 2 ] · [( 3 PQ ) 2 − ( PQ ) 2 ] + ( P Q ) 4 + ( Q P ) 4 − 10 = 0. ( 28 ) 2. A Set of Main Results In this section, we state and prove a set of six new theta-function identities which depict inter-relationships among q -product identities and the multivariate R -functions. Theorem 3. Each of the following relationships holds true : R 1 R 3 R 5 R 15 = ( R 3 R 5 R 1 R 15 ) 2 − ( q 2 R 1 R 15 R 3 R 5 ) 2 + ( 3 qR 3 R 5 R 1 R 15 } + ( 3 q 3 R 1 R 15 R 3 R 5 ) − ( 5 q 4 R 5 R 15 R 1 R 3 ) (29) and ( R 1 R 5 R 3 R 15 ) 2 = ( R 3 R 5 R 1 R 15 ) 3 − ( q 2 R 1 R 15 R 3 R 5 ) 3 − ( 5 q 2 R 3 R 5 R 1 R 15 ) + ( 5 q 4 R 1 R 15 R 3 R 5 ) + 5 q 5 ( R 1 R 15 R 3 R 5 ) 2 + 5 q ( R 3 R 5 R 1 R 15 ) 2 − ( 3 q 3 R 3 R 15 R 1 R 5 ) 2 (30) 6 Mathematics 2020 , 8 , 918 Equations ( 29 ) and ( 30 ) give inter-relationships between R 1 , R 3 , R 5 and R 15 ( R 1 R 7 q 2 R 3 R 21 ) 3 · ( q 12 [ R 1 R 21 ] 8 [ R 3 R 7 ] 8 − 1 ) = 1 q 5 ( [ R 1 ] 3 R 7 [ R 3 ] 3 R 21 ) 2 ( 7 − [ R 1 ] 4 q [ R 3 ] 4 ) + ( q [ R 1 ] 3 R 21 [ R 3 ] 3 R 7 ) 2 ( [ R 1 ] 4 q [ R 3 ] 4 − 3 ) − 14 q 3 ( [ R 1 ] 5 R 7 [ R 3 ] 5 R 21 ) · ( q 6 [ R 1 R 21 ] 4 [ R 3 R 7 ] 4 − 1 ) − 27 q 6 ( R 1 R 21 R 3 R 7 ) 4 + 21 q ( R 1 R 3 ) 4 + 21 q 5 ( [ R 1 ] 2 R 21 [ R 3 ] 2 R 7 ) 4 − 7 q 2 ( R 1 R 3 ) 8 (31) Equation ( 31 ) gives inter-relationships between R 1 , R 3 , R 7 , and R 21 ( R 1 R 3 ) 2 = ( q 1 2 R 1 R 6 R 2 R 3 ) 2 + ( ( 3 q ) 1 2 R 3 R 1 ) 2 + ( R 2 R 3 R 1 R 6 ) 2 (32) Equation ( 32 ) gives inter-relationships between R 1 , R 2 , R 3 , and R 6 ( R α R 2 ( q 6 ; q 6 ) ∞ R 6 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ ) 2 = ( ( 3 q ) 1 2 R α ( q 6 ; q 6 ) ∞ ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ ) 2 + ( ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ R α ( q 6 ; q 6 ) ∞ ) 2 + ( q 1 2 R 6 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ R α R 2 ( q 6 ; q 6 ) ∞ ) 2 (33) Equation ( 33 ) gives inter-relationships between R 2 , R 6 , and R α . Furthermore , it is asserted that ( R 3 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ R 1 R α ( q 6 ; q 6 ) ∞ ) 4 + ( R 1 R α ( q 6 ; q 6 ) ∞ R 3 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ ) 4 + [ ( R 3 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ R 1 R α ( q 6 ; q 6 ) ∞ ) 2 − ( R 1 R α ( q 6 ; q 6 ) ∞ R 3 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ ) 2 ] · [ ( 3 q 1 2 R α R 3 ( q 6 ; q 6 ) ∞ R 1 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ ) 2 − ( R 1 ( q 2 ; q 2 ) ∞ ( − q 3 ; q 6 ) ∞ q 1 2 R α R 3 ( q 6 ; q 6 ) ∞ ) 2 ] − 10 = 0. (34) Equation ( 34 ) gives inter-relationships between R 1 , R 3 and R α It is assumed that each member of the assertions ( 29 ) to ( 34 ) exists. Proof. First of all, in order to prove the assertion (29) of Theorem 3, we apply the identity (9) (with q replaced by q 3 , q 5 q 15 ) under the given precondition of result (23). Thus, by using (20) and (21), and, after some simplifications, we get the values for P and Q as follows: P = ψ ( q ) q 1 2 ψ ( q 5 ) = R 1 q 1 2 R 5 ( 35 ) and Q = ψ ( q 3 ) q 3 2 ψ ( q 15 ) = R 3 q 3 2 R 15 ( 36 ) Now, upon substituting from these last results (35) and (36) into (23), if we rearrange the terms and use some algebraic manipulations, we are led to the first assertion (29) of Theorem 3. Secondly, we prove the second relationship (30) of Theorem 3. Indeed, if we first apply the identity (9) (with q replaced by q 3 , q 5 and q 15 ) under the given precondition of the assertion (24), and then make use of (20) and (21), after some simplifications, the following values for P and Q would follow: 7