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Pentti Haukkanen Pauliina Ilmonen Ayse Nalli Juha Sillanpää 《Linear and Multilinear Algebra》2013,61(5):599-616
A divisor d ∈ ?+ of n ∈ ?+ is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM. 相似文献
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Pentti Haukkanen 《Linear and Multilinear Algebra》2013,61(3):301-309
Considering lower closed sets as closed sets on a preposet (P, ≤), we obtain an Alexandroff topology on P. Then order preserving functions are continuous functions. In this article we investigate order preserving properties (and thus continuity properties) of integer-valued arithmetical functions under the usual divisibility relation of integers and power GCD matrices under the divisibility relation of integer matrices. 相似文献
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Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Qh(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers. 相似文献
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Some analogues of smith's determinant 总被引:1,自引:0,他引:1
We calculate the determinants of the greatest common divisor (GCD) and the least common multiple (LCM) matrices associated with an arithmetical function on gcd-closed and lcm-closed sets. We also consider some unitary analogues of these determinants. 相似文献
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Pentti Haukkanen 《Aequationes Mathematicae》2017,91(3):527-536
It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with \(f(1)\ne 0\) which satisfy the arithmetical equation \( f(mn)f((m,n)) = f(m)f(n)g((m, n)) \) for all positive integers m, n with \(m,n \in A(mn)\), where A is a regular convolution and g is an A-multiplicative function. Euler’s totient function \(\phi _A\) with respect to A is an example satisfying this equation. 相似文献
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An integer a is said to be regular (modr) if there exists an integer x such that a 2 x≡a (mod r). In this paper we introduce an analogue of Ramanujan’s sum with respect to regular integers (modr) and show that this analogue possesses properties similar to those of the usual Ramanujan’s sum. 相似文献
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Pentti Haukkanen 《数学研究及应用》1997,17(4):519-523
Recently, L.C.Hsu and Wang Jun generated new combinatorial number theoreticfunctions serving as generalizations of Euler′s totient. In this paper we form an extensive class of generalized Euler totients by translating the most general counting functions of Hsu and Wang on integers to the setting of Narkiewicz′s regular convolution. This class casts in the same framework various famous generalizations of Euler′s totient, such as Cohen′s totient, Jordan′s totient, Klee′s totient, Schemmel′s totient,Stevens′s to tient, the unitary analogue of Euler′s to tient and Euler′s to tient with respect to Narkiew icz′s regular convo lution. 相似文献