Related papers: Two identities involving Cohen-Ramanujan expansion…

An arithmetical function $f$ is said to be even (mod r) if f(n)=f((n,r)) for all n\in\Z^+, where (n, r) is the greatest common divisor of n and r. We adopt a linear algebraic approach to show that the Discrete Fourier Transform of an even…

Let $R(q)$ be the Rogers-Ramanujan continued fraction. We give different proofs of two complementary relations for $R(q)$ given by Ramanujan and proved by Watson and Ramanathan. Our proofs only use product expansions for classical Jacobi…

If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of…

We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also…

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the…

A numerical scheme is developed for the evaluation of Abramowitz functions $J_n$ in the right half of the complex plane. For $n=-1,\, \ldots,\, 2$, the scheme utilizes series expansions for $|z|<1$ and asymptotic expansions for $|z|>R$ with…

The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$,…

Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\,(\text{mod } n)$ for $b,n\in\mathbb{Z}$. By $(a,b)_s$, we mean the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define…

Using a general $q$-summation formula, we derive a generating function for the $q$-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q$-Hahn polynomials. A new proof of the orthogonality…

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical…

The polynomial Ramanujan sum was first introduced by Carlitz [7], and a generalized version by Cohen [10]. In this paper, we study the arithmetical and analytic properties of these sums, derive various fundamental identities, such as H…

We consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicative functions as extensions of the class of multiplicative functions. We apply these concepts to Ramanujan's sum and its analogue with respect to…

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula…

In this self-contained short note, we introduce the new definition of Good Ramanujan Expansion, say G.R.E., for a fixed arithmetic function $F$, building upon a good decay of its coefficients $G$; this, gains $\log-$powers w.r.t. the…

We prove a simple formula for the main value of $r$-even functions and give applications of it. Considering the generalized Ramanujan sums $c_A(n,r)$ involving regular systems $A$ of divisors we show that it is not possible to develop a…

We generalize Ramanujan's expansions of the fractional-power Euler functions (q^{1/5})_{\infty} = [ J_1 - q^{1/5} + q^{2/5} J_2 ](q^5)_{\infty} and (q^{1/7})_{\infty} = [ J_1 + q^{1/7} J_2 - q^{2/7} + q^{5/7} J_3 ] (q^7)_{\infty} to…

We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and…

We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions…

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\phi$. We derived the Menon-type identity $\sum\limits_{\substack{m=1\\(m.n^s)_s=1}}^{n^s} (m-1,n^s)_s=\Phi_s(n^s)\tau_s(n^s)$ in Czechoslovak Math.…

Having a function $f$ and a set of functionals $\{\mathcal{C}_{n}\}$, $c_n^f \equiv \mathcal{C}_n \left(f\right)$, one can interpret function approximation very generally as a construction of some function $\mathcal{A}_{N}^{f}$ such that…