Analysis of a generalized Lebesgue identity in Ramanujan’s Lost Notebook

We analyze a two-parameter q-series identity in Ramanujan’s Lost Notebook that generalizes the product part of the fundamental one-parameter Lebesgue identity. From reformulations of this two-parameter identity, we deduce new partition theorems including variants of the Gauss triangular number identity and Euler’s pentagonal number theorem. We discuss connections with a partial theta identity of Ramanujan and with several classical results such as those of Sylvester and Göllnitz–Gordon.     

Combinatorial Proofs of Identities in Ramanujan’s Lost Notebook Associated with the Rogers-Fine Identity and False Theta Functions

Ramanujans lost notebook contains several identities arising from the Rogers-Fine identity and/or Rogers false theta functions. Combinatorial proofs for many of these identities are given.AMS Subject Classification: 05A17, 11P81.     

Ramanujan’s Master Theorem

S. Ramanujan introduced a technique, known as Ramanujan??s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan??s Master Theorem is discussed.     

“Algebraical oddities” in Ramanujan’s lost notebook

In his lost notebook, Ramanujan makes certain claims, described by Hardy as ??algebraical oddities??. We give proofs of two of these.     

On Plouffe’s Ramanujan identities

Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry’s constant given by Ramanujan:

$\zeta(3)=\frac{7\pi^{3}}{180}-2\sum_{n=1}^{\infty}\frac {1}{n^{3}(e^{2\pi n}-1)}.$

Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe’s identities as corollaries. The resemblance of the general relation to the structure of theta functions and modular forms is briefly sketched.

     

Ramanujan’s inverse elliptic arc approximation

We suggest a continued fraction origin to Ramanujan’s approximation to $(\frac{a-b}{a+b})^{2}$ in terms of the arc length of an ellipse with semiaxes a and b.     

Analogues of Ramanujan’s partition identities

Ramanujan discovered that $$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$ where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for $p_{[c^{l}d^{m}]}(n)$ .     

A generalization of Knopp’s Observation on Ramanujan’s tau-function

The Ramanujan Journal - We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan’s tau-function.     

An algorithmic approach to Ramanujan’s congruences

In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?_d|M?_n=1^￥(1-q^dn)^r_d=?_n=0^￥a(n)qⁿ\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).     

A refinement of Ramanujan’s factorial approximation

We present a new proof of the monotonicity of the correction term θ _n in Ramanujan’s refinement of Stirling’s formula. Moreover we prove that θ _n is concave.     

Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(q ⁱ ) (i=1,2,3,4), Q(q ⁱ ) (i=1,2,3), and R(q ⁱ ) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ? of any three-element subset of {P(q),P(q ²),P(q ³),P(q ⁴)} and of any two-element subset of {Q(q),Q(q ²),Q(q ³)} and {R(q),R(q ²),R(q ³)}, respectively. For all the results we use some expressions of $P(q^{i_{1}}), Q(q^{i_{2}}) $ , and $R(q^{i_{3}}) $ in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.     

Combinatorial proofs of an identity from Ramanujan’s lost notebook and its variations

We give two combinatorial proofs and partition-theoretic interpretations of an identity from Ramanujan’s lost notebook. We prove a special case of the identity using the involution principle. We then extend this into a direct proof of the full identity using a generalization of the involution principle. We also show that the identity can be rewritten into a modified form that we prove bijectively. This fits the identity into Pak’s duality of partition identities proven using the involution principle and partition identities proven bijectively. The original identity was first proven algebraically by Andrews as a consequence of an identity of Rogers’ and combinatorially by Kim, while the modified form of the identity generalizes an identity recently found by Andrews and Warnaar related to the product of partial theta functions.     

Ramanujan’s modular equations of degree 5

We provide alternative derivations of theta function identities associated with modular equations of degree 5. We then use the identities to derive the corresponding modular equations.     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.     

Arithmetic properties arising from Ramanujan’s theta functions

We prove some interesting arithmetic properties of theta function identities that are analogous to q-series identities obtained by Michael D. Hirschhorn. In addition, we find infinite family of congruences modulo powers of 2 for representations of a non-negative integer n as \(\triangle _1+4\triangle _2\) and \(\triangle +k\square \).     

Ramanujan’s cubic transformation and generalized modular equation

We study the quotient of hypergeometric functionsμ_a~*(r)=π/2sin(πa) F(a,1-a;1;1-r~3)/F(a,1-a;1;r~3),r∈(0,1)in the theory of Ramanujan's generalized modular equation for a ∈(0,1/2],and find an infinite product formula for μ_(1/3)~*(r) by use of the properties of μ_a~*(r) and Ramanujan's cubic transformation.Besides,a new cubic transformation formula of hypergeometric function is given,which complements the Ramanujan's cubic transformation.     

Ramanujan’s enigmatic formula for the harmonic series

We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as “enigmatic”, for the harmonic series.