Theorems Stated by Ramanujan (III) : Theorems on Teansformation of Series and Integrals

Preece, C. T.

doi:10.1112/jlms/s1-3.4.274

Cited by 4 publications

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“…Note, however, that in a letter to Hardy [, p. XXVI, VI (8)] Ramanujan stated the related formula

\sum_{k = 1}^{\infty} \frac{k^{4 m}}{{prefixsinh}^{2} (k π)} = - \frac{1}{2 π} B_{4 m} - \frac{4 m}{π} \sum_{k = 1}^{\infty} \frac{k^{4 m - 1}}{1 - {normale}^{2 π k}}, m \in N,

corresponding to the special case of

s = 1

but with

j = 0

, being outside the range of validity of Theorem . This formula was first proven by Preece using the residue theorem. Remark The (first) Bernoulli numbers are defined recursively via, see ,

B_{0} = 1, B_{1} = - \frac{1}{2}, B_{ℓ} = - \sum_{k = 0}^{ℓ - 1} (\frac{0pt}{ℓ}) \frac{B_{k}}{ℓ - k + 1} for ℓ \geq 2 .

Hence

B_{0} = 1, B_{1} = - 1 / 2, B_{2} = 1 / 6, B_{4} = - 1 / 30, B_{6} = 1 / 42, B_{8} = - 1 / 30, \dots,

while the Bernoulli numbers with odd index larger than one vanish, i.e.…”

Section: Resultsmentioning

confidence: 95%

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Nonlinear differential identities for cnoidal waves

Leitner

Mikikits-Leitner

2014

Mathematische Nachrichten

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This article presents a family of nonlinear differential identities for the spatially periodic function us(x), which is essentially the Jacobian elliptic function cn 2(z;m(s)) with one non‐trivial parameter s>0. More precisely, we show that this function us fulfills equations of the form us(α)us(β)(x)=∑n=02+α+βbα,β(n)us(n)(x)+cα,β,for all α,β∈double-struckN0. We give explicit expressions for the coefficients bα,β(n) and cα,β for given s. Moreover, we show that for any s the set of functions {1,usa,us′,us′′,⋯} constitutes a basis for L2(0,2π). By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well‐known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.

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“…Note, however, that in a letter to Hardy [, p. XXVI, VI (8)] Ramanujan stated the related formula

\sum_{k = 1}^{\infty} \frac{k^{4 m}}{{prefixsinh}^{2} (k π)} = - \frac{1}{2 π} B_{4 m} - \frac{4 m}{π} \sum_{k = 1}^{\infty} \frac{k^{4 m - 1}}{1 - {normale}^{2 π k}}, m \in N,

corresponding to the special case of

s = 1

but with

j = 0

, being outside the range of validity of Theorem . This formula was first proven by Preece using the residue theorem. Remark The (first) Bernoulli numbers are defined recursively via, see ,

B_{0} = 1, B_{1} = - \frac{1}{2}, B_{ℓ} = - \sum_{k = 0}^{ℓ - 1} (\frac{0pt}{ℓ}) \frac{B_{k}}{ℓ - k + 1} for ℓ \geq 2 .

Hence

B_{0} = 1, B_{1} = - 1 / 2, B_{2} = 1 / 6, B_{4} = - 1 / 30, B_{6} = 1 / 42, B_{8} = - 1 / 30, \dots,

while the Bernoulli numbers with odd index larger than one vanish, i.e.…”

Section: Resultsmentioning

confidence: 95%

“…corresponding to the special case of s = 1 but with j = 0, being outside the range of validity of Theorem 2.1. This formula was first proven by Preece [31] using the residue theorem.…”

Section: )mentioning

confidence: 91%

Nonlinear differential identities for cnoidal waves

Leitner

Mikikits-Leitner

2014

Mathematische Nachrichten

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Some elegant approximations and asymptotic formulas of Ramanujan

Berndt

Evans

1991

Journal of Computational and Applied Mathematics

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Series transformations and integrals involving the Riemann Ξ-function

Dixit

2010

Journal of Mathematical Analysis and Applications

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The transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar are unified, in the sense that all these formulas come from the same source, namely, a general formula involving an integral of Riemann's Ξ -function. We give proofs of all of these transformation formulas using the theory of Mellin transforms and the residue theorem. Our study includes new extensions of the formulas of Koshliakov and Ferrar through their connection with integrals involving the Riemann Ξ -function.

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Theorems Stated by Ramanujan (III) : Theorems on Teansformation of Series and Integrals

Cited by 4 publications

References 0 publications

Nonlinear differential identities for cnoidal waves

Nonlinear differential identities for cnoidal waves

Some elegant approximations and asymptotic formulas of Ramanujan

Series transformations and integrals involving the Riemann Ξ-function

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