Evaluation of the convolution sums ∑ al + bm = n lσ(l)σ(m) withab≤ 9

Yoon Kyung Park

Displaying similar documents to “Evaluation of the convolution sums ∑ al + bm = n lσ(l)σ(m) withab≤ 9”

Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Ebénézer Ntienjem (2017)

Open Mathematics

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The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{matrix} \sum_{\binom{(l, m) \in ℕ_{0}^{2}}{α l + β m = n}} σ (l) σ (m), \end{matrix}$ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive...

On the incomplete gamma function and the neutrix convolution

Brian Fisher, Biljana Jolevska-Tuneska, Arpad Takači (2003)

Mathematica Bohemica

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The incomplete Gamma function $γ (α, x)$ and its associated functions $γ (α, x_{+})$ and $γ (α, x_{-})$ are defined as locally summable functions on the real line and some convolutions and neutrix convolutions of these functions and the functions $x^{r}$ and $x_{-}^{r}$ are then found.

The Neutrix Convolution Product x -r,- ○ x μ,+

Brian Fisher, Emin Özcag (1991)

Publications de l'Institut Mathématique

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On a convolution type integral I

S. R. Yadava (1972)

Matematički Vesnik

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On a class of arithmetic convolutions involving arbitrary sets of integers.

Tóth, László (2002)

Mathematica Pannonica

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Compact convolution operators between $L_{p} (G)$ -spaces

G. Crombez, W. Govaerts (1978)

Colloquium Mathematicae

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A characterization of α-convolutions

J. Kucharczak (1973)

Colloquium Mathematicae

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A characterization of a class of convolutions

Kazimierz Urbanik (1967)

Colloquium Mathematicum

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On a class of $ψ$ -convolutions characterized by the identical equation

Jean-Louis Nicolas, Varanasi Sitaramaiah (2002)

Journal de théorie des nombres de Bordeaux

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The identical equation for multiplicative functions established by R. Vaidyanathaswamy in the case of Dirichlet convolution in 1931 has been generalized to multiplicativity preserving $ψ$ -convolutions satisfying certain conditions (cf. [7]) which can be called as Lehmer-Narkiewicz convolutions for some reasons. In this paper we prove the converse.

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.