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The harmonic logarithms and the binomial formula

Steven Roman
1993 Journal of combinatorial theory. Series A  
LOGARITHMIC BINOMIAL FORMULA Now let us turn to the logarithmic binomial formula.  ...  Thus, we see that the logarithmic binomial formula is indeed a generalization of the classical binomial formulas (1) and (2) .  ... 
doi:10.1016/0097-3165(93)90030-c fatcat:ssme3iqmsfhx3o5bthxvthzdya

The logarithmic fibbinomial formula [article]

A.K.Kwasniewski
2004 arXiv   pre-print
Roman logarithmic binomial formula analogue has been found .  ...  It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.  ...  "Recent Contributions to the calculus of Finite Differences: a Survay" Lecture Notes in Pure and Appl.  ... 
arXiv:math/0406258v1 fatcat:erire5eminacvjqh2pxcixxtzi

Logarithms of a binomial series: A Stirling number approach [article]

Helmut Prodinger
2018 arXiv   pre-print
The p-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers.  ...  Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.  ...  Then C(z) = 1 + u, and, by the Lagrange inversion formula [7] , u m = n≥m m n 2n n − m z n for m ≥ 1. For m = 0 the formula is still true when taking a limit.  ... 
arXiv:1812.11805v1 fatcat:tv3mhgm7rbab5do3txvxznqzpe

Logarithms of a binomial series: A Stirling number approach

Helmut Prodinger
2019 Ars Mathematica Contemporanea  
The p-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers.  ...  Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.  ...  Introduction Knuth [6, 7] proposed the exciting formula (log C(z)) 2 = n≥1 2n n (H 2n−1 − H n ) z n n , where with the generating function of Catalan numbers and harmonic numbers.  ... 
doi:10.26493/1855-3974.1901.987 fatcat:63o6nhw5sneo5jhyk6ccz4lsaa

Formal power series of logarithmic type

Daniel E Loeb, Gian-Carlo Rota
1989 Advances in Mathematics  
In terms of the harmonic numbers, we define the harmonic logarithm of order t and degree n by the formula l*;'(x) = Y c (-l)k (t)k Q'(log x)'-k.  ...  We stress the fact that this formula is an identity in the logarithmic algebra, and not just an asymptotic formula.  ... 
doi:10.1016/0001-8708(89)90079-0 fatcat:2pzpz5km2bg2nig3ypfk2xfwk4

Page 139 of Mathematics Magazine Vol. 8, Issue 6 [page]

1934 Mathematics Magazine  
factorials and their logarithms up to 11, Bernouilliis numbers and their logarithms up to B and Euler’s numbers and their logarithms up to E;, as well as formulas of differentiation and integral tables  ...  The integrals of trigonometric functions are preceded by trigo- nometric identities, Euler’s formula, DeMoivre’s formula and appli- cations, formulas for plane triangles, trigonometric series, and the  ... 

Page 139 of Mathematics Magazine Vol. 8, Issue 6 [page]

1934 Mathematics Magazine  
factorials and their logarithms up to 11, Bernouilli’s numbers and their logarithms up to B and Euler’s numbers and their logarithms up to E;, as well as formulas of differentiation and integral tables  ...  The integrals of trigonometric functions are preceded by trigo- nometric identities, Euler’s formula, DeMoivre’s formula and appli- cations, formulas for plane triangles, trigonometric series, and the  ... 

The iterated logarithmic algebra II: Sheffer sequences [article]

Daniel E. Loeb
1995 arXiv   pre-print
This leads to several examples including Stirling's formula and a logarithmic version of the Euler-MacLaurin summation formula.  ...  An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials.  ...  We stress the fact that this formula is an identity in the logarithmic algebra, and not just an asymptotic formula.  ... 
arXiv:math/9502220v1 fatcat:cz63v2ezyjgstczugjdb7w63ey

The Interated Logarithmic Algebra

Daniel E Loeb
1991 Advances in Mathematics  
a formula for the action of a product of two Artinian operators on the harmonic logarithm.  ...  Is there a simple formula expressing any monomial 1' or product of harmonic logarithms I:(x) 1:(x) in terms of harmonic logarithms?  ...  All of the formulas for composition and inversions of series (in particular, the various versions of the LaGrange 212 DANIEL E. LOEB inversion formula) are consequences of the following theorem.  ... 
doi:10.1016/0001-8708(91)90041-5 fatcat:xikyd5jc5zhmzhtwwljjz2o43q

Page 5 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
Wen Li Chen (PRC-SWNU; Chongqing) 94i:05007 05A40 33E20 39A70 Roman, Steven (1-CAS3; Fullerton, CA) The harmonic logarithms and the binomial formula. J. Combin. Theory Ser.  ...  Harmonic logarithms are introduced as the natural logarithmic analogue of powers of x, since they obey a logarithmic binomial theorem A!)(x +a) = Dol" A, (a)x*.  ... 

Page 4142 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews  
A “logarithmic binomial theorem” is derived as well as explicit formulas for the harmonic logarithms.  ...  Bundschuh (D-KOLN) 93h:05014 05A40 Roman, Steven (1-CAS3) The logarithmic binomial formula. Amer. Math. Monthly 99 (1992), no. 7, 641-648.  ... 

Sums of series involving central binomial coefficients & harmonic numbers [article]

Amrik Singh Nimbran
2019 arXiv   pre-print
An elegant sum involving ζ(2) and two other nice sums appear in the last section.  ...  This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers.  ...  Acknowledgement: The author is thankful to Prof K. N. Boyadziev for suggesting explicit use of generating functions, and to Prof P. Levrie for referring to a formula in [14] .  ... 
arXiv:1806.03998v2 fatcat:gj7pcd2ppfd5dphtpdpzhhdpdm

Page 3165 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews  
They introduce another basis of L, whose ele- ments are called the harmonic logarithms, and denoted by A‘).  ...  Pleasantly, it turns out that any fact about poly- nomials that can be stated using the operators x and D has a logarithmic extension involving the operators o and D, where the harmonic logarithms play  ... 

Expressions for the Entropy of Binomial-Type Distributions [article]

Mahdi Cheraghchi
2018 arXiv   pre-print
As a result, we derive series expansions and integral representations of the entropy for several fundamental distributions, including the Poisson, binomial, beta-binomial, negative binomial, and hypergeometric  ...  We develop a general method for computing logarithmic and log-gamma expectations of distributions.  ...  Acknowledgement The author thanks an anonymous reviewer for comments on related works [Kne98, JS99, CGKK13] .  ... 
arXiv:1708.06394v4 fatcat:glu5koyecraejekd2m6qfodxgy

Dances between continuous and discrete: Euler's summation formula [article]

David J. Pengelley
2019 arXiv   pre-print
I will show in his own words Euler's idea for deriving his summation formula, and how he applied the formula to the sum of reciprocal squares and other situations, e.g., large factorials and binomial coefficients  ...  He applied it to estimate harmonic series partial sums, the gamma constant, and sums of logarithms, thereby calculating large factorials (Stirling's series) with ease.  ...  He also applied the formula to harmonic partial sums and the related gamma constant, and to sums of logarithms, thereby approximating large factorials (Stirling's asymptotic approximation) and binomial  ... 
arXiv:1912.03527v1 fatcat:vccaxdxgaraylltdqpng63pd6u
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