1 On a class of harmonic-like numbers ________________________________
Definition
The classical harmonic numbers (see for instance [1] for a definition and their main properties) H n = / + / + ..... + / n possess the integral representation from which a "natural" generalisation seems to be, for any complex number a : In effect a n - t n = (a - t)(a n - 1 + a n - 2 t +.....+ a n - p t p - 1 + ....t n - 1 ) , and, through a term by term integration, we obtain Then if we set a = 1 in this relation we find the usual harmonic numbers. Hence the label "harmonic-like" given to the numbers H n (a) . An application to analysis
We defer the study of the general number H n (a) to a further note, and instead we focus on the particular case H n ( ½ ) , to find an application to the expansion into series of some special functions. To this end we need to transform the integral representation (1) . By some obvious substitutions of variable in (1) we get: npaaa pnnn aH n (1)dta ta taH nn n dzz nndyy nndtn zyn ttH n dt ttH n n
10 11 and finally Now we consider the integral sine function, classically defined [2] as or equivalently If we multiply both sides of this last relation by the first derivative of Si(z) , i.e., (sin z)/z , then But since , this leads to Now by using the MacLaurin expansion of the cosu functions under the integral sign (with u = z(1 - x) and u = z (1 + x) successively), then the expression (2) of the numbers H n ( ½ ) , and integrating the result with regard to z , we finally obtain a power series expansion of the square of the integral sine function, valid for any complex number z , which contains H n ( ½ ) coefficients of only even subscripts. On the other hand, by the same technique as above, we obtain (2)dxx nnn xxn H .0 sin)( dxz xxzSi .10 sin)( dxxzxzSi .10 sinsin2²)( dxxz zxzzSidzd .10 )1(cos)1(cos²)( dxxz xzxzzSidzd .)2( 2)()()!( 11²)( z n212n2nnzSi n H n .)2( 12)()!1( 11)(cos2 z n212nnzSiz n H n which contains H n ( ½ ) coefficients of only odd subscripts. If we replace the trigonometric functions sint and cost by their hyperbolic counterparts sht and cht , the same techniques as above lead us to similar series expansions. Other property of H n (½) In passing, we notice that the numbers H n ( ½ ) display a remarkable relationship with the inverses of binomial coefficients , for we have the following identity, several times independently proved ([3], [4] and [5]), and even generalised ([6]) According to our definition the right-hand side of this relation is equal to (n + 1)H n+1 ( ½ ) . Therefore , for n > 1 we have the simple identity: _______________________ REFERENCES Ronald L. Graham, Donald E. Knuth and Oren Patashnik,
Concrete Mathematics , Addison-Wesley Publishing Company (1989) pp 258-268 2.
Milton Abramowitz and Irene A. Stegun,
Hanbook of mathematical functions , Dover Publications Inc., New York (1970) p 231 3.
Tor B. Staver, On Summasjion av Potenser av Binomialkoeffisientene,
Norsk Matematisk Tidsskrift (1947) pp 97-103 4. Andrew M. Rockett, Sums of the Inverses of Binomial Coefficients,
The Fibonacci Quarterly (1981) pp 433-437 5.
Juan Pla, The Inverses of Binomial Coefficients Revisited,
The Fibonacci Quarterly (1997) pp 342-345 6.
Zhizheng Zhang and Haitao Song, A Generalization of an identity involving the inverses of binomial coefficients,
Tamkang Journal of Mathematics , (Autumn 2008) pp 219-226 (see http://journals.math.tku.edu.tw/index.php/TKJM/article/download/14/34 ) Juan PLA, 315 rue de Belleville 75019 Paris (France) .11111 k2knk2nnnp pn 1 .111)( np pn 1n21 nn