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The harmonic logarithms and the binomial formula

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) . ...

##
###
The logarithmic fibbinomial formula
[article]

2004
*
arXiv
*
pre-print

Roman

arXiv:math/0406258v1
fatcat:erire5eminacvjqh2pxcixxtzi
*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. ...##
###
Logarithms of a binomial series: A Stirling number approach
[article]

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. ...

##
###
Logarithms of a binomial series: A Stirling number approach

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. ...

##
###
Formal power series of logarithmic type

1989
*
Advances in Mathematics
*

In terms of

doi:10.1016/0001-8708(89)90079-0
fatcat:2pzpz5km2bg2nig3ypfk2xfwk4
*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*. ...##
###
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]

1995
*
arXiv
*
pre-print

This leads to several examples including Stirling's

arXiv:math/9502220v1
fatcat:cz63v2ezyjgstczugjdb7w63ey
*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*. ...##
###
The Interated Logarithmic Algebra

1991
*
Advances in Mathematics
*

a

doi:10.1016/0001-8708(91)90041-5
fatcat:xikyd5jc5zhmzhtwwljjz2o43q
*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. ...##
###
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]

2019
*
arXiv
*
pre-print

An elegant sum involving ζ(2)

arXiv:1806.03998v2
fatcat:gj7pcd2ppfd5dphtpdpzhhdpdm
*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] . ...##
###
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]

2018
*
arXiv
*
pre-print

As a result, we derive series expansions

arXiv:1708.06394v4
fatcat:glu5koyecraejekd2m6qfodxgy
*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] . ...##
###
Dances between continuous and discrete: Euler's summation formula
[article]

2019
*
arXiv
*
pre-print

I will show in his own words Euler's idea for deriving his summation

arXiv:1912.03527v1
fatcat:vccaxdxgaraylltdqpng63pd6u
*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*...
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