Generalized harmonic sum
WebThe H n, r generalized harmonic number is defined as: H n, r = ∑ k = 1 n 1 k r I'm interested in the growth of H n, r as a function of n, for a fixed r ∈ [ 0, 1]. For r > 1, H n, r = O ( 1) (as a function of n ). For r = 1, H n, 1 = O ( log n) . For r = 0, H n, 0 = n. How does H n, r grow for intermediate values of r? summation asymptotics WebJan 8, 2016 · You can't find a general formula. All you can do is the use the standard asymptotic formula for the harmonic sum H n = ∑ k = 1 n 1 k = ln n + γ + 1 2 n − 1 12 n …
Generalized harmonic sum
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WebNov 1, 2011 · The generalized harmonic numbers H n ( s) of order s are defined by ( cf . [1]; see also [2] and [3, p. 156]) (1.1) H n ( s) ≔ ∑ j = 1 n 1 j s ( n ∈ N; s ∈ C) and (1.2) H n ≔ H n ( 1) = ∑ j = 1 n 1 j ( n ∈ N) are the harmonic numbers. WebThe main purpose of this paper is to define multiple alternative q-harmonic numbers, Hnk;q and multi-generalized q-hyperharmonic numbers of order r, Hnrk;q by using q-multiple zeta star values (q-MZSVs). We obtain some finite sum identities and give some applications of them for certain combinations of q-multiple polylogarithms …
WebMay 27, 2016 · This question must have been asked, it's just very hard to search for such questions. I'm looking for the cleanest method I can find for getting a closed form formula for $\sum_{i=1}^n i^k$ WebMay 18, 2024 · The generalised harmonic number of order m of n is. H n, m = ∑ k = 1 n 1 k m. For example, the harmonic numbers are H n, 1, and H ∞, 2 = π 2 6. These are …
WebApr 13, 2024 · Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas - We confirm two conjectural congruences of Sun in Sun (Int J Math … Webproperties of the generalized harmonic sum P n k=m 1=k k, where m; m+1:::; nare positive integers. At the end of this note we point out a connection between the arithmetic properties of harmonic sums and the distribution of primes as well as potential generalizations of harmonic numbers. 1. Introduction
WebJul 20, 2015 · Harmonic numbers. The nth harmonic number, H n, is the sum of the reciprocals of the integers up to and including n.For example, H 4 = 1 + 1/2 + 1/3 + 1/4 = 25/12.. Here’s a curious fact about harmonic numbers, known as Wolstenholme’s theorem:. For a prime p > 3, the numerator of H p-1 is divisible by p 2.. The example above shows …
WebSep 15, 2010 · We define generalized harmonic number sums (4) S j ( b, k) ≡ ∑ n = 1 ∞ n j H n ( k) b n + 1, b > 1, wherein we also allow b = −1. For k = 1 we may use the well-known generating function for harmonic numbers, and we thereby obtain various logarithmic sums. More interesting is the k = 2 case connected with the dilogarithmic function Li 2. paint software free download for windows 8WebIn this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very … paint software free download windows 10Webgeneralized harmonic mean and generalized geometric mean are more robust than those based on the L 1 metric and L 2 metric. Figure 1. The retrieval accuracy of four metrics on image database; for Cauchy metric, a 5; and for 1 st-type generalized harmonic mean, p 1.8. To compare the image retrieval results, we query the image sugar factory hard rock tampaWebSep 16, 2024 · This paper is concerned with the combinatorial identities of the harmonic and the hyperharmonic Fibonacci numbers. By using the symmetric algorithm, we get some identities which improve the usual results and generalize known equations. Moreover, with the help of concept of Riordan array, we obtain the generating functions for these … sugar factory hard rockWebJan 9, 2016 · You can't find a general formula. All you can do is the use the standard asymptotic formula for the harmonic sum H n = ∑ k = 1 n 1 k = ln n + γ + 1 2 n − 1 12 n 2 + 1 120 n 4 +... where γ ≈ 0.5772156649 is the Euler–Mascheroni constant. Your sum would be H b − H a − 1. Share Cite Follow edited Jul 8, 2013 at 20:57 Mahdi Khosravi 3,969 24 50 paint software online freeWebOct 15, 2015 · It can be shown that this volume is equal to the trace of a compact self-adjoint operator. We provide an explicit expression for the kernel of this operator in … paint software free onlineWebMar 28, 2011 · Abstract: Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. … paint software free download