scholarly journals An identity for a class of arithmetical functions of two variables

1988 ◽  
Vol 11 (2) ◽  
pp. 351-354
Author(s):  
J. Chidambaraswamy ◽  
P. V. Krishnaiah
Keyword(s):  
Positive Integer ◽  
Wide Class ◽  
Arithmetical Functions ◽  
Functions Of Two Variables ◽  
Special Cases ◽  
Ramanujan’S Sum ◽  
Ramanujan's Sum

For a positive integerr, letr∗denote the quotient ofrby its largest squarefree divisor(1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1)   or   0according asr∗|nor not whereC(n,r)is the well known Ramanujan's sum. In this paper, using a different method, we generalize(∗)to a wide class of arithmetical functions of2variables and deduce as special cases(∗)and similar formulae for several generalizations of Ramanujan''s sum.

scholarly journals An identity for a class of arithmetical functions of several variables

1993 ◽  
Vol 16 (2) ◽  
pp. 355-358
Author(s):  
Pentti Haukkanen
Keyword(s):  
Wide Class ◽  
Trigonometric Sum ◽  
Arithmetical Functions ◽  
Functions Of Two Variables ◽  
Several Variables ◽  
Special Cases

Johnson [1] evaluated the sum∑d|n|C(d;r)|, whereC(n;r)denotes Ramanujan's trigonometric sum. This evaluation has been generalized to a wide class of arithmetical functions of two variables. In this paper, we generalize this evaluation to a wide class of arithmetical functions of several variables and deduce as special cases the previous evaluations.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

Reciprocity in Ramanujan's Sum

1986 ◽  
Vol 59 (4) ◽  
pp. 216 ◽  
Author(s):  
Kenneth R. Johnson
Keyword(s):  
Ramanujan’S Sum ◽  
Ramanujan's Sum

Some Properties of Generalized Euler Numbers

1981 ◽  
Vol 33 (3) ◽  
pp. 606-617 ◽  
Author(s):  
D. J. Leeming ◽  
R. A. Macleod
Keyword(s):  
Positive Integer ◽  
Roots Of Unity ◽  
Euler Numbers ◽  
Special Cases ◽  
Image Position

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by(1.1)where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is(1.2)Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

On the Davison Convolution of Arithmetical Functions

1989 ◽  
Vol 32 (4) ◽  
pp. 467-473 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Positive Integer ◽  
Arithmetical Functions ◽  
Positive Divisor ◽  
Complex Valued ◽  
Ordered Pairs

AbstractThe Davison convolution of arithmetical functions f and g is defined by where K is a complex-valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. In this paper we shall consider the arithmetical equations f(r) = g, f(r) = fg, f o g = h in f and the congruence (f o g)(n) = 0 (mod n), where f(r) is the iterate of f with respect to the Davison convolution.

Generalized Ramanujan's sum

1979 ◽  
Vol 10 (1) ◽  
pp. 71-87 ◽  
Author(s):  
J. Chidambaraswamy
Keyword(s):  
Ramanujan’S Sum ◽  
Ramanujan's Sum

A note on two-variable Lommel matrix functions

2018 ◽  
Vol 11 (03) ◽  
pp. 1850041 ◽  
Author(s):  
Ayman Shehata
Keyword(s):  
Matrix Functions ◽  
Functions Of Two Variables ◽  
Special Cases

The aim of the present work is to develop a pair of Lommel matrix functions of two variables suggested by the Bessel matrix functions and some of their properties are studied to be special cases of our results.

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