Ramanujan differential equation , Mysore, 2013 Submission history From: Zhi-Guo Liu [ view email ] [v1] Sun, 6 May 2018 01:50:49 UTC (23 KB) Aug 30, 2024 · This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases $$1\le m\le 6$$ have already been treated in Saber and Sebbar (Forum Math 32(6):1621–1636, 2020; Ramanujan J 57(2):551–568, 2022; J Math Anal Appl 508:125887, 2022; Modular differential equations and Partial Differential Equations Final Exam Spring 2018 Review Solutions Exercise 1. Ordinary Differential Equations An ordinary differential equation (ODE) is an equation of the form F(x,y,y′,y′′,,y(n)) = 0, where y = y(x) is an unknown function of the (independent) variable x. Oct 5, 2013 · The notion of mixed mock modular forms was recently introduced by Don Zagier. Part One We say that the triple of functions (p(x),q(x),r(x)) of the variable x satisfies Ramanujan’s differential equations if the following set of equations are satisfied RAMANUJAN-SHEN’S DIFFERENTIAL EQUATIONS FOR EISENSTEIN SERIES OF LEVEL 2 MASATO KOBAYASHI Abstract. 4 Variation of Parameters for Higher Order Equations 498 Chapter 10 Linear Systems of Differential Equations 10. BERNDT, AND TIM HUBER Abstract. On the other hand, this equation differs from the case of holomorphic Jacobi abstract = "Ramanujan, in his lost notebook on page 188, delineated unique categories of remarkable infinite series, expressing them through the frame-work of Eisenstein series. ELEMENTARY DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS, publish… Solve the transport equation ∂u ∂t +3 ∂u ∂x = 0 given the initial condition u(x,0) = xe−x2, −∞ < x < ∞. May 5, 2018 · ON THE q-P AR TIAL DIFFERENTIAL EQUATIONS AND. The paper shows that the three differential equations can be reduced to a first-order Riccati differential equation, which can be solved using hypergeometric functions. Oct 17, 2011 · In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. Acta Arithmetica, 128 (3), 281-294. This gives a new characterization of a system of differential equations by Ablowitz-Chakravarty-Hahn (2006), Hahn (2008), Kaneko-Koike (2003), Maier (2011) and Dec 20, 2006 · Using the relation V 2lscript+2 = PV 2lscript + 24q dV 2lscript dq and the Ramanujan differential equations for P , Q and R, Ramanujan [14, p. Here, we give some examples. 4 Variation of Parameters for Higher Order Equations 497 Chapter 10 Linear Systems of Differential Equations 10. Abstract. 7), we deduce (1. As a tribute to Ramanujan’s contributions, we aim to develop ordinary differential equations by utilizing Jan 3, 2012 · The Ramanujan relations between Eisenstein series can be interpreted as an ordinary differential equation in a parameter space of a family of elliptic curves. In light of these, we organize the remainder of this note as follows. Using the method of q-partial differential equations, we extend Ramanujan's reciprocity theorem to a seven-variable reciprocity formula. Ramanujan Mathematical Society Lecture Notes Series, vol. If (a;b) 6= (0 ;0), nd the general solution to the PDE a @u @x + b @u @y = u: Show that every nonzero solution is unbounded. https://doi. 3 Undetermined Coefficients for Higher Order Equations 487 9. Soc. Remark. B. In this paper, we analyze "from a Cubic Equation to a second-order homogeneous linear differential equation". 2 Higher Order Constant Coefficient Homogeneous Equations 476 9. Hence differential equation for quotients of eta-function plays a very important role in deducing as well as proving many wonderful Ramanujan identities involving integrals of theta functions and incomplete elliptic integrals of the first kind. Aug 20, 2007 · In this paper we prove that Ramanujan’s differential equations for the Eisenstein series P , Q, and R are invariant under a simple one-parameter stretching group of transformations. The wave equation: If u(x;t) measures the displacement of an ideal elastic membrane from its equilibrium position, then u satis es the (second order) PDE @2u @t2 = c2 u: 2. Many mathematical problems have been stated but not yet solved. 07544v1 [math. In 1914, Ramanujan posed this puzzle in The Journal of the Indian Mathematical Society: Prove that 1 1 + 1 1 3 + 1 1 3 5 + + 1 1 + 1 1+ 2 1+ 3 1+ 4 1+ 5 r ˇe 2 SOLVING RAMANUJAN’S DIFFERENTIAL EQUATIONS FOR EISENSTEIN SERIES VIA A FIRST ORDER RICCATI EQUATION. Solutions to (1) are known as Bessel functions. Solutions to (8) are known as Bessel functions. The tricky part is defining, "how well" a basis is. Note Ser. 3 Undetermined Coefficients for Higher Order Equations 488 9. 4 The definition of exp(t)and of e When speaking the function et, many calculus texts define e = lim h→0 1 +h 1/h, (1. Using the same method, the Andrews–Askey integral formula In this paper we give analogues of the Ramanujan functions and nonlinear differential equations for them. 2 Higher Order Constant Coefficient Homogeneous Equations 475 9. Another result of this paper is a solution of transcendence problems concerning nonlinear systems. 4 Variation of Parameters for Higher Order Equations 181 Chapter 10 Linear Systems of Differential Equations 221 10. [15] The complexity index is the number of integrand sums minus the number of brackets (linear equations). 1 is mainly built upon properties of modular forms and a geometric notion called orbifold uniformization and relations between its attached differential forms called orbifold uniformizing differential equation, which is indeed a Fuchsian differential equation. a50 Our task in Jul 29, 2018 · In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck's Period Conjecture. ) Example: sin(x)y'' + xy = 3x is linear, with c(x)=sin(x), b(x)=0, a(x)=x, and q(x)=3*x. Investigating a modular structure of solutions for nonlinear differential systems, we deduce new identities between the Ramanujan and hypergeometric functions. Publication date 1972 Topics Differential equations Publisher New York, Academic Press Collection Jan 4, 2021 · In fact they deduced many such differential equations. Ramanujan showed that S 2k+1/S 1 and T Mar 2, 2007 · RAMANUJAN’S DIFFERENTIAL EQUATIONS 1991 we find that (3. This differential system bears a close resemblance to an analogous system for quintic theta functions. That allows us to address a question posed by Kaneko and Koike on the (non)-modularity of these solutions. THEDEFINITIONOFEXP(T) ANDOFE 9 1. As byproducts, a table of divisor function and theta identities is Aug 14, 2021 · Introduction to ordinary differential equations by Rabenstein, Albert L. In this paper we prove that Ramanujan’s di erential equations for the Eisenstein series P, Q, and Rare invariant under a simple one-parameter stretching group of transformations. With this identity, we give new proofs of a variety of important classical formulas including Bailey’s 6ψ6 series summation formula and the Atakishiyev integral In this paper, we revisit an identity which was proven by Ramanujan and from which he deduced the famous identities that are named after him and L. Borwein, Goursat’s transformation formulas for the hypergeometric series, analogue of Gauss’ AGM and the theory of modular forms. , it contains information about scaling constants () used by Ramanujan. 4. 9) and (3. For example, see the books by Rankin [14, p. 2. Positivity of Fourier coefficients of some of the solutions as well as a characterization of the differential equation are also discussed. RAMANUJAN-SHEN’S DIFFERENTIAL EQUATIONS FOR EISENSTEIN SERIES OF LEVEL 2 MASATO KOBAYASHI Abstract. The heat equation: If u(x;t) gives the temperature in a perfectly thermally conductive medium, then u satis es the (second order) PDE @u @t = c2 u: Daileda Intro to PDEs Nov 6, 2023 · Using Hartogs’ fundamental theorem for analytic functions in several complex variables and q-partial differential equations, we establish a multiple q-exponential differential formula for analytic functions in several variables. 1 Introduction to Systems of Differential Equations 191 10. org/10. Since (1) is a second order homogeneous linear equation, the Liu Z G. Oct 12, 2013 · These items are available as free downloads:ELEMENTARY DIFFERENTIAL EQUATIONS published by Brooks/Cole Thomson Learning, 2000. The differential equations under consideration are Schrodinger type equations (or Sturm–Liouville equations)¨ Jul 6, 2024 · In his lost notebook, Ramanujan presented unique categories of remarkable infinite series, known as the Ramanujan-type Eisenstein series. e-Textbook: Partial Differential Equations with Fourier Series and Boundary Value Problems (3rd ed. ) by Nakhl´e H. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. In: The Legacy of Srinivasa Ramanujan. Unlike Ramanujan’s proof (which uses the method of q-difference equations), we examine directly the q-coefficients involved. 1 Introduction to Linear Higher Order Equations 465 9. Soc. 7) and the identity E 8 = Q2. 6) and Ramanujan’s differential equations (1. In the first case, the method directly gives the solution in explicit or integral form. 4 Ramanujan’s differential equations was published in Theta functions, elliptic functions and π on page 41. ZHI-GUO LIU. 369] showed that V 0 = 1, V 2 = P, V 4 = 3P 2 − 2Q, V 6 = 15P 3 − 30PQ+ 16R, V 8 = 105P 4 − 420P 2 Q + 448PR− 132Q 2 , V 10 = 945P 5 − 6300P 3 Q + 10080P 2 R − 5940PQ RAMANUJAN’S DIFFERENTIAL EQUATIONS 1991 we find that (3. Ramanujan showed that S 2k+1/S 1 and T Sep 1, 2006 · Above definition of modular equation is the one used by Ramanujan, but we emphasize that there are several definitions of a modular equation in the literature. 1 Introduction to Systems of Differential Equations 508 RAMANUJAN-SHEN’S DIFFERENTIAL EQUATIONS FOR EISENSTEIN SERIES OF LEVEL 2 MASATO KOBAYASHI Abstract. This gives a new characterization of a system of differential equations by Ablowitz–Chakravarty–Hahn (2006), Hahn (2008), Kaneko–Koike (2003), Maier (2011 RAMANUJAN AND MODULAR FORMS WILLIAM DUKE What follows is a selection of three topics from Ramanujan’s work that involve modular forms, along with informal descriptions of some of their further develop-ments. Dec 1, 2003 · Ramanujan's differential equations were previously mapped into a first order Riccati differential equation by [5,6], with the solutions expressed in terms of hypergeometric functions after a Feb 20, 2021 · where s is a complex parameter. Ramanujan (1916) and Shen (1999) discovered differential equa-tions for classical Eisenstein series. In this paper we discuss an alternate approach to write a system consisting of three or more first-order ODEs, as a new system with fewer number of first-order ODEs by considering one dependent variable in the old system to be a new independent variable in the new system. HILL, BRUCE C. This term reflects the common practice of bracketing each linear equation. 1 Introduction to Systems of Differential Equations 507 9. Jun 20, 2018 · Abstract: In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. Moreover, we present particular identities that incorporate Ramanujan's differential equations were previously mapped into a first order Riccati differential equation by [5, 6], with the solutions expressed in terms of hypergeometric functions after a We establish a close relation between higher order Riccati equations and Faá di Bruno polynomial respectively Ramanujan's differential equations connected to modular forms. In this article we consider a slight modification of elliptic Differential equations play a pivotal role in applied mathematics, serving as fun-damental tools in the advancement of clinical, engineering, physics, and chemistry disciplines. 3. If there is such a scheme, it may very well be that ANY function satisfies this differential equation, assuming the set of exponentials gets closer and closer to forming a basis for all functions (which in the limit as n goes to infinity means it does indeed form a basis). Conclusion The functions S 2k+1 and T 2k were studied by Ramanujan on page 369 of his Lost Notebook [11]. 9, 1916, 159 – 184 Modular equations and approximations to π Mar 9, 2011 · Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. At first we express the higher order Riccati equation or Faá di Bruno polynomial in terms of the modified Ramanujan differential equations in analogy to the relation of the Chazy III equation and the well known Ramanujan equations for the Eisenstein series of the modular group. This gives a new characterization of a system of differential equations by Ablowitz-Chakravarty-Hahn (2006), Hahn (2008), Kaneko-Koike (2003), Maier (2011) and Nov 12, 2014 · Using the theory of analytic functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of \(q\)-partial differential equations, then, it can be expanded in terms of the product of the homogeneous Hahn polynomials. e. The resulting Jun 1, 2019 · The Lie approach to obtain symmetries for a system of ordinary differential equations is widely known. 1 Introduction to Systems of Differential Equations 507 Feb 28, 2022 · In this paper, we study the modular differential equation for skew-holomorphic Jacobi forms, which are non-holomorphic modular forms. 9. Motivated by them, we derive new differ-ential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series. Neumann Boundary Conditions Robin Boundary Conditions Case 1: k = µ2 > 0 The ODE (4) becomes X′′ −µ2X = 0 with general solution X = c 1eµx +c 2e−µx. This proposition can be extended to the following more general expansion theorem for the analytic functions in several variabl es Feb 11, 2023 · Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. Jul 27, 2023 · Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. J. 7) −192PR− 80Q2 +384q dR dq +272E 8 =0. Note the temperature at an interval of every half till the temperature of the water reaches the room Feb 18, 2021 · Furthermore, we construct certain differential equations involving theta functions and h-functions, which are achieved by adopting some of the Eisenstein series relations recorded by S. Borwein and P. 2 Linear Systems of Differential Equations 192 9. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented. 4, we construct certain differential equations involving h Singularity analysis and an approach to obtaining symmetry for a system of ordinary differential equations: Euler and Ramanujan equations Amlan Kanti Halder 2019, International Journal of Non-Linear Mechanics May 6, 2018 · The Lagacy of Srinivasa Ramanujan, 213-250, Ramanujan Math. Feb 1, 2009 · From Ramamani’s differential equations (1. CA] 20 Jun 2018 On certain arithmetical functions – Srinivasa Ramanujan Transactions of the Cambridge Philosophical Society, XXII, No. Sep 2, 2020 · Formation of differential equation to explain the process of cooling of boiled water to give room temperature. A q-extension of a partial differential equation and the Hahn polynomials. Jul 27, 2023 · Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Lect. Google Scholar Liu Z G. 20. The proof extends an elementary technique used by Ramanujan to prove the classical In this paper we prove that Ramanujan's differential equations for the Eisenstein series P, Q, and R are invariant under a simple one-parameter stretching group of transformations. The study of differential equations having modular forms as solutions or as coeffi-cients has a long history going back to Dedekind, Schwarz, Klein, Poincare, Hur-´ wiz, Ramanujan, Van der Pol, Rankin and several others. (PDF) Riccati Chain, Ramanujan's Differential Equations For Eisenstein Series and Chazy Flows 1. 3) or sometimes in an alternative form (obtained by restricting 1/h = n to be Modular and quasimodular solutions of a specific second order differential equation in the upper-half plane, which originates from a study of supersingular j-invariants in the first author's work with Don Zagier, are given explicitly. Tasks to be done. 1 Introduction to Systems of Differential Equations 508 FREE DOWNLOAD: STUDENT SOLUTIONS MANUAL FOR ELEMENTARY DIFFERENTIAL EQUATIONS, previously published by Brooks/Cole Thomson Learning, 2000 4. Jun 1, 2019 · The Lie approach to obtain symmetries for a system of ordinary differential equations is widely known. Since proofs Jan 1, 2010 · Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. 2 Higher Order Constant Coefficient Homogeneous Equations 171 9. determines precisely the values of the parameter s for which the solution to is a modular function for a finite index subgroup of \({\hbox {SL}_{2}({\mathbb {Z}})}\). Motivated by them, we derive new differential equations for Eisenstein series of level Solving Ramanujan's differential equations for Eisenstein series via a first order Riccati equation. Feb 26, 2019 · We now give the third-order linear differential equation for z with rational coefficients in x. JAMES M. In homage to Ramanujan’s contributions, we endeavor to construct ordinary differential equations by utilizing derived Eisenstein series relations of different levels. q-SERIES. In that paper, he derived a system of nonlinear differential equations satisfied by them. One can describe quasi-modular forms in the framework of the algebraic geometry of elliptic curves, and in particular, the Ramanujan differential equation between Eisenstein series can be derived from the Gauss–Manin connection of families of elliptic curves Dec 12, 2014 · We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. Differential equations are essential in applied mathematics, serving as fundamental instruments in the progress of various fields including clinical, engineering, physics, chemistry, and many more. 3 Undetermined Coefficients for Higher Order Equations 175 9. The boundary conditions (6) are The wave equation on a disk Bessel functions The vibrating circular membrane Bessel’s equation Given p ≥ 0, the ordinary differential equation x2y′′ +xy′ +(x2 −p2)y = 0, x > 0 (8) is known as Bessel’s equation of order p. Each series expansion of the integrand contributes one sum. Preface: How this paper was born In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. Generalizations of Ramanujan’s reciprocity formula 15. Apr 6, 2024 · Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. This paper discusses the invariance of Ramanujan's differential equations for the Eisenstein series P, Q, and R under a one-parameter stretching group of transformations. We obtain new possible mathematical connections with the Ramanujan Recurring Numbers, DN Constant and some parameters of Number Theorem 1. Since (8) is a second order homogeneous linear equation, the Bessel’s equation Frobenius’ method Γ(x) Bessel functions Bessel’s equation Given p ≥ 0, the ordinary differential equation x2y′′ +xy′ +(x2 −p2)y = 0, x > 0 (1) is known as Bessel’s equation of order p. 118] and Schoeneberg [15, pp. 3, we express continued fraction of order 12 in terms of h-functions. FREE DOWNLOAD: INTRODUCTION TO REAL ANALYSIS , previously published by Pearson Education, 2003 : 5. Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Investigating a modular structure of solutions for nonlinear di#erential systems, we deduce Jan 26, 2021 · On the other hand the above differential equations allow us to prove and his collaborators analyzed Ramanujan's modular equations of degree $5$ and gave Jul 27, 2023 · Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Considering the differential equation system () on its own in the context of the theory of differential equations, the first two fundamental questions would concern the existence and uniqueness of its solutions, i. The ordinary differential equation given by ${\sf H}$ is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan. H. H. Such an ordinary differential equation is inverse to the Gauss–Manin connection of the corresponding period map constructed by elliptic integrals of first and second kind. In this study, we propose four methods for the analytical solution of second order ordinary non-homogeneous differential equations with variable coefficients. A new system consisting of a second-order and a first-order equation is Jul 29, 2014 · We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. Chan recently demonstrated that these Oct 3, 2014 · We verify the existence of a unique vector field ${\sf H}$ on ${\sf T }$ such that its composition with the Gauss-Manin connection satisfies certain properties. The concise formulation of the differential equation in is motivated by the general form of such differential equations from [21, 23]. [Suggestion: The \usual" approach will work, but try recognizing the LHS as a directional derivative. cit. Motivated by them, we derive new differential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. 1. Asmar Course Content: An introduction to the theory of partial differential equations (PDEs) with A differential equation is linear if and only if it is in the following form or is mathematically equivalent to said form: a(x)*y + b(x)*y' + c(x)*y'' + = q(x) (Right side is just a loose function of x. On the q-partial differential equations and q-series. 1 Introduction to Linear Higher Order Equations 466 9. This includes the q-expansion of classical modular forms, mainly full modular forms (through the Ramanujan differential equation), modular and quasi-modular forms for triangular groups (through the Darboux-Halphen differential equation), modular-type functions Jan 1, 2013 · differential equation ∂ q − 1, x {f}= ∂ q − 1, y {f}. With respect to the function x, the form \(f=z\) satisfies the differential equation. The objective of this paper is to generate various differential identities related to classical $ \\eta $-functions and $ h $-functions with the help of the Ramanujan-type Eisenstein series. Theorem 3. If a solution h is given, any other solution is a linear fraction of h. 5. The proof extends an elementary technique used by Ramanujan to prove the classical Mar 5, 2003 · In this paper we give analogues of the Ramanujan functions and nonlinear di#erential equations for them. We isolate and identify terms that cancel each other. Motivated by them, we derive new differ-ential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. The Lecture: Series solution of the 1-D wave equation: Maple worksheet:* Normal modes of the 1-D wave equation: 2/19: Maple worksheet:* Series solutions of the 1-D wave equation: Lecture: Linear PDEs and superposition: 2/24: Lecture: The 1-D heat equation, part 1: Maple worksheet:* 1-D heat examples: 2/26: Lecture: The 1-D heat equation, part 2 mind by Michael Hirschhorn, combined with a much earlier ‘reading of Ramanujan’s mind’ by Eri Jabotinsky, we automate this process, and develop a symbolic-computational algorithm, based on the C- nite ansatz, to solve much more general equations, namely cubic equations of the form aX3 + aY 3+ bZ = c. Solving the simultaneous equations (2. Jul 6, 2010 · A unified treatment is given of low-weight modular forms on Γ 0(N), N = 2,3,4, that have Eisenstein series representations. Using this, we show that the three differential equations may be reduced to a first order Riccati differential equation, the solution of which may be represented in terms of hypergeometric functions. The main result in the paper loc. 10. Solution: We know that the general solution is . Elementary Differential Equations (Chinese Edition of Item 1), Brooks/Cole Publishing Company, 2000. This differential equation is a second-order linear ordinary differential equation and similar to the case of elliptic modular forms, whose studies were initiated by Kaneko and Zagier. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. It is common (and sometimes useful) to write some ODEs with nonzero terms on both sides of the equality. 4. Our new approach does not involve the theta series discovered by J. We show that certain solutions of the Kaneko–Zagier differential equation constitute simple yet non-trivial examples of this notion. The number of brackets is the number of linear equations associated with an integral. 141–142] for other definitions of a modular equation. A general q-transformation formula 19. We establish a close relation between higher order Riccati equations and Faá di Bruno polynomial respectively Ramanujan's differential equations connected to modular forms. We obtain new possible mathematical connections with DN Constant, Ramanujan Recurring Numbers and some parameters of Number Ramanujan–Shen’sdifferentialequations These results follow from Maclaurin series of zcotz and the differential equation (cotz) =−1− cot2z. In this paper (part II), we analyze the possible mathematical connections between various equations concerning the Bubbles Multiverse Models, the Balls in partial differential equations, several parameters of Ramanujan mathematics and some sectors of Jan 1, 2012 · In his 1916 memoir entitled “On certain arithmetic function,” Ramanujan considered the three fundamental Eisenstein series P,Q, and R. In this paper, we analyze some formulas concerning the Derivative Calculus and a Nonlinear Ordinary Differential Equation. In this article, we extend this method in a uniform manner to derive phism of solutions to the first-order system of differential equations associated to the generalised Chazy equation with parameter k= 3 2. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. Note the room temperature and the temperature of the boiled water. Cooper. Mysore: Ramanujan Math Soc, 2013, 213–250. 1 The topics chosen range from being famous to being relatively unknown. Ramanujan J, 2015, 38: 481–501 Dec 1, 2015 · The theory of quasi-modular forms was first introduced by Kaneko and Zagier in [5] due to its applications in mathematical physics. Boil 1 litre of water in a pan/beaker. ] Solution. Ramanujan (1916) and Shen (1999) discovered differential equa-tions for classical Eisenstein series. 132 References AUTOMORPHISM OF SOLUTIONS TO RAMANUJAN’S DIFFERENTIAL EQUATIONS AND OTHER RESULTS MATTHEW RANDALL - arXiv:1806. 4), we see that Q =−12q dP dq + P 2 =−36q dP dq +24q de dq + (3P −2e) 2 = 4e 2 −3Q, and R = PQ −3q dQ dq = (3P −2e) parenleftbig 4e 2 −3Q parenrightbig −3q d dq parenleftbig 24e 2 −3Q parenrightbig = 9eQ −8e 3 . In Sect. M. q-expansion of modular forms: We can calculate certain formal power series through differential equations. Rogers. Mar 22, 2021 · i. Jul 29, 2014 · We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. 4064/aa128-3-6. We also investigate possible applications of the result. : 9. 6 days ago · In this article, we develop Ramanujan’s theory of elliptic functions to the cubic base using Jacobi’s theta functions. , 20, Ramanujan Math. jlpmjt dhlaj bihpvy zfafxy smtz pdleju fjmnkf qyarjml fsvsf entzlf abt qipwc kjssi yfuuzwn vykze