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Some another remarks on the generalization of Bernoulli and Euler numbers.

Short review of classical approaches

Bernoulli numbers were first introduced by Jacques Bernoulli (1654-1705), in the second part of his treatise published in 1713, Ars conjectandi , at the time, Bernoulli numbers were used for writing the infinite series expansions of hyperbolic and trigonometric functions. Van den berg was the first to discuss finding recurrence formulae for the Bernoulli numbers with arbitrary sized gaps (1881) [9]. Ramanujan showed how gaps of size 7 could be found, and explicitly wrote out the recursion for gaps, of size 6 [10]. Lehmer in 1934 extended these methods to Euler numbers, Genocchi numbers, and Lucas numbers (1934) [9], and calculated the 196-th Bernoulli numbers. Bernoulli polynomials play an important role in various expansions and approximation formulas which are useful both in analytic theory of numbers and in classical and numerical analysis. These polynomials can be defined by various methods depending on the applications. In particular, six approaches to the theory of Bernoulli polynomials are known, these are associated with the names of J. Bernoulli, L. Euler, P. E. Appel, A. Hurwitz, E. Lucas and D. H. Lehmer. Also Apostol and Qiu-Ming Luo defined new generalizations of Bernoulli polynomials that we have used in this paper.

[sections]1. Generalized Raabe multiplication theorem

For a real or complex parameter [alpha] , the higher order Bernoulli polynomials [B.sup.([alpha]).sub.n] (x) and the higher order Euler polynomials E([alpha]) n (x), each of degree n in x as well as in [alpha], are defined by the following generating functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That the explicit formula for [B.sup.([alpha]).sub.n] (x) and E([alpha]) n (x) are

[B.sup.([alpha]).sub.n] = [n.summation over (k=1)] [sigma](n, k)[[alpha].sup.k],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

respectively. See [1], [2].

Moreover, the higher order Bernoulli numbers [B.sup.([alpha]).sub.n] and higher order Euler numbers [E.sup.([alpha]).sub.n] are defined by

[(t/[e.sup.t] - 1).sup.[alpha]] = [[infinity].summation over (n=0)] [B.sup.([alpha]).sub.n] [t.sup.n]/n! ,

and

[(2/[e.sup.t] - 1).sup.[alpha]] = [[infinity].summation over (n=0)] [E.sup.([alpha]).sub.n] [t.sup.n]/n!

respectively.

Clearly, for all nonnegative integers n, the classical Bernoulli and Euler polynomials, [B.sub.n](x) and [E.sub.n](x) are given by [B.sub.n](x) := [B.sup.(1).sub.n] (x) and [E.sub.n](x) := [E.sup.(1).sub.n] (x):

That the classical Bernoulli polynomials [B.sub.n](x) and Euler polynomials [E.sub.n](x) are defined through the generating functions

[te.sup.xt]/[e.sup.t] - 1 = [[infinity].summation over (n=0)] [B.sub.n](x) [t.sup.n]/n!,

and

[2e.sup.xt]/[e.sup.t] - 1 = [[infinity].summation over (n=0)] [E.sub.n](x) [t.sup.n]/n!.

The explicit formulas for [B.sub.n](x) and [E.sub.n](x), are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [B.sub.k] := [B.sub.k](0) is the k-th Bernoulli number and [E.sub.k] := [E.sub.k](1) is the k-th Euler number. The Bernoulli numbers may also be calculated from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also the Bernoulli numbers are given by the double sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Bernoulli numbers satisfy the sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At first we introduce necessary definitions about this matter.

Definition 1.1.([2]) Let a, b > 0, a [not equal to] b, the generalized Bernoulli numbers [B.sub.n](a, b) are defined by

t/[b.sup.t] - [a.sup.t] = [[infinity].summation over (n=0)] [B.sub.n](a, b)/n! [t.sup.n],

where |t| < 2[pi]/|ln b-ln a|.

Definition 1.2. Let a, b > 0, a [not equal to] b, we define the generalized Bernoulli polynomials as

[te.sup.xt]/[b.sup.t] - [a.sup.t] = [[infinity].summation over (n=0)] [B.sub.n](x; a, b)/n! [t.sup.n],

where |t| < 2[pi] |ln b-ln a|.

Definition 1.3. For positive numbers a, b, the generalized Euler numbers [E.sub.k](a, b) are defined by

2/[b.sup.2t] - [a.sup.2t] = [[infinity].summation over (n=0)] [E.sub.k](a, b)/k! [t.sup.k].

Definition 1.4. For any given positive numbers a, b and x [member of] R, the generalized Euler polynomials [E.sub.k](x; a, b) are defined by

[2e.sup.xt]/[b.sup.t] - [a.sup.t] = [[infinity].summation over (n=0)] [E.sub.k](x; a, b)/k! [t.sup.k].

Theorem 1.1.([12]) For positive numbers a, b, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 1.1. In special case if we set b = e, a = 1, y = 0, then we obtain

[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the G-S.Cheon formula. (See [1] for detail)

The term Bernoulli polynomials was used first in 1851 by Raabe [10] in connection with the following multiplication theorem

1/m [m-1.summation over (k=0)] [B.sub.n](x + k/m) = [m.sup.-n][B.sub.n](mx).

Here we give an analogues formula for generalized Bernoulli numbers.

Theorem 1.2. Let x, y, a, b [member of] C (Complex numbers) so we have the following identity

1/m [m-1.summation over (k=0)] [B.sub.n] (x + k/m ln a + m - k + 1/m ln b) = [m.sup.-n][B.sub.n](mx):

Proof. Let us expand the function

[[infinity].summation over (n=0)] [B.sub.n](x, a, b)/n! [t.sup.n] = t/[b.sup.t] - at ext, b [not equal to] a: (1:1)

In powers of x and t and collect the coefficients of [t.sup.n]/n! as a polynomial [[PSI].sub.n](x, a, b) of degree n in x :

F(x, t, a, b) = [[infinity].summation over (n=0)] [[PSI].sub.n](x, a, b) [t.sup.n]n! .

Suppose

[[PSI].sub.n](x, a, b) = [A.sup.(n).sub.0] [x.sup.n] + [A.sup.(n).sub.1] [x.sup.n-1] + ... + [A.sup.(n).sub.n].

That

[A.sup.(n).sub.i] := [A.sup.(n).sub.i] (a, b).

If we replace x by 1/y and t by ty in (1.1) we get

F(1/y, ty, a, b) = ty/[b.sup.ty] - [a.sup.ty] [e.sup.t] = [[infinity].summation over (n=0)] yn[[PSI].sub.n](1/y, a, b) [t.sup.n]/n!.

Letting y tend to zero we obtain

1/ln b - ln a [e.sup.t] = [[infinity].summation over (n=0)] [A.sup.(n).sub.0] [t.sup.n]/n!.

Hence, [A.sup.(n).sub.0] = 1/ln b-ln a and hence [[PSI].sub.n](x, a, b) is monic.

If in (1.1) we replace x by x+ln [a.sup.k/m] [b.sup.m-k+1/m] and sum over k and divide the result by m we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

If, instead, we replace in (1.1) x by mx and t by t/m we obtain

F(mx, t/m, a, b) = t/m[e.sup.xt]/[b.sup.t/m] - [a.sup.t/m] = [[infinity].summation over (n=0)] 1/[m.sup.n] [[psi].sub.n](mx, a, b) [t.sup.n]/n!. (3.1)

Identifying coefficients of [t.sup.n]/n! in (2.1), (3.1) we conclude that [[PSI].sub.n](x, a, b) satisfies the functional equation

1/m [m-1.summation over (k=0)] [[PSI].sub.n] (x + k/m ln a + m - k + 1/m ln b) = [m.sup.-n][[PSI].sub.n](mx).

Because [[PSI].sub.n](x, a, b) is monic therefore proof is complete.

Now according to a next lemma we give a representation matrix for [B.sup.-1.sub.n] and [E.sup.-1.sub.n] .

[section]2. Matrix representation of [B.sup.-1.sub.n] and [E.sup.-1.sub.n]

Lemma 2.1.([3]) we have

[([[infinity].summation over (n=0)] [a.sub.n][x.sup.n]).sup.-1] = 1/[a.sub.0] + [[infinity].summation over (n=1)] [(-1).sup.n][x.sup.n]/n![a.sup.n+1.sub.0] [G.sub.n],

(let [a.sub.0] [not equal to] 0),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now according to previous lemma and because

[([[infinity].summation over (n=0)] [B.sub.n]/n! [x.sup.n]).sup.-1] = [[infinity].summation over (n=0)] [B.sup.(-1).sub.n]/n! [x.sup.n].

So if we set an = [B.sub.n]/n! then

[[infinity].summation over (n=0) [B.sup.(-1).sub.n] [x.sup.n] = 1 + [[infinity].summation over (n=1)] [(-1).sup.n]/n] [G.sub.n][x.sup.n]/n!.

Now if [G.sup.*.sub.n] := [G.sub.n], n [greater than or equal to] 1 and [G.sup.*.sub.0] 0 = 1 so [B.sup.(-1).sub.n] = [(-1).sup.n][G.sup.*.sub.n].

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [G.sup.*.sub.0] = 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[sections]3. Euler Maclaurin summation Formula for [B.sub.n,[alpha]]

Let [g.sub.[alpha]](z) := [2.sup.[alpha]][GAMMA]([alpha] + 1) [J.sub.[alpha]](z)/[z.sup.[alpha]],

where

[J.sub.[alpha]](z) = [[infinity].summation over (k=0)] [(-1).sup.k][z.sup.2k+[alpha]]/ [2.sup.2k+[alpha]]k!-([GAMMA] + k + 1)

is the Bessel function of the first kind order [alpha].

The function [J.sub.[alpha]](z)/[z.sup.[alpha]] is an even entire function of exponential type one, we assume that [alpha] is not a negative integer. The zeros [j.sub.k] = [j.sub.k]([alpha]) of J[alpha](z) z[alpha] may then be ordered in such a way that [j.sub.-k] = [-j.sub.k] and 0 < |[j.sub.1]| [less than or equal to] |[j.sub.2]| [less than or equal to] ... . We define a sequence of polynomials [B.sub.n],[sigma](x) by the generating function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

We call the polynomials [B.sub.n,[sigma]](x) the [alpha]-Bernoulli polynomials and [B.sub.n,[sigma]](0) =: [B.sub.n,[sigma]] the [alpha]-Bernoulli numbers.

To easily we see [B.sub.0,[sigma]](x) = 1,[B.sub.1,[sigma]](x) = x - 1/2 ,[B.sub.2,[sigma]](x) = [(x - 1/2).sup.2] - 1/8([alpha]+1) , ... . And also to easily of (1.3) we can prove

[B'.sub.n,[alpha]](x) = n[B.sub.n-1],[sigma](x), n = 1, 2, 3, ... .

[B.sub.n,[sigma]](1 - x) = (-1)n[B.sub.n,[sigma]](x), n = 1, 2, 3, ... .

in particular [B.sub.n],[sigma](1) = (-1)nBn,[sigma]: (see [4])

In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus.

In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler-Maclaurin formula is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where the symbol indicates that the right-hand side is a so-called asymptotic series for the left-hand side. This means that if we take the first n terms in the sum on the right-hand side , the error in approximating the left-hand side by that sum is at most on the order of the (n+l)st term.

Now we will find a same formula for generalized Bernoulli numbers [B.sub.n],[sigma].

Theorem 3.1.([6]) Let f be a real function with continuous (2k)th derivative. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[n.summation over (i=1)] f(i) = [S.sub.k] - [R.sub.k],

where the error term is

[R.sub.k] = [[integral].sup.n.sub.1] [f.sup.(2k)](t)[B.sub.2k] ({t})/(2k)! dt

with [B.sub.2k](t) the Bernoulli polynomial and {t} = t - [t] the fractional part of t.

Now we consider Euler Maclaurin summation Formula for [B.sub.n,[sigma]], The technique employs repeated integration by formula

[B'.sub.n,[alpha]](x) = n[B.sub.n-1,[alpha]], n = 1, 2, 3, ...

to create new derivatives. We start with

[[integral].sup.1.sub.0] f(x)dx = [[integral].sup.1.sub.0] f(x)[B.sub.0,[alpha]](x)dx. (2.3)

Because [B'.sub.1,[alpha]](x) = [B.sub.0,[alpha]](x) = 1 substituting [B'.sub.1,[alpha]](x) in (2.3) and integrating by parts, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Again we have [B.sub.1,[alpha]](x) = 1/2[B'.sub.2,[alpha]](x) and integrating by parts

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the relation

[B.sub.n,[alpha]](1) = (-1)n[B.sub.n,[alpha]](0) = (-1)n[B.sub.n,[alpha]], (n = 0, 1, 2, 3, ...)

And continuing this process , we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

This is the generalization of Euler-maclaurin integration formula , it assume that the function f(x) has the required derivatives. The rang of integration in (2.2) my be shifted [0,1] to [1,2] by replacing f(x) by f(x + 1). Adding such results up to [n - 1, n],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The terms

1/2f(0) + f(1) + f(2) + ... + f(n - 1) + 1/2f(n)

appear exactly as in trapezoidal integration or quadrature .

[sections]4. Identity for Apostol Bernoulli numbers

Definition 4.1. The Apostol Bernoulli numbers fin([greater than or equal to]) are defined by means of the generating functions

t/[lambda][e.sup.t] - 1 = [[infinity].summation over (n=0)] [[beta].sub.n] ([lambda]) [t.sup.n]/n! , |t + log [lambda]| < 2[pi].

That [[beta].sub.n]([greater than or equal to]) is called Apostol Bernoulli numbers.

Lemma 4.1.([6]) Suppose that |x| < 1 so we have

[[infinity].summation over (k=0)] f(k)[x.sup.k] = -[[infinity].summation over (m=0)] [f.sup.(m-1)](0)/m! [[beta].sub.m](x), |x| < 1.

Now according to pervious lemma we give one identity for Apostol Bernoulli numbers.

Corollary 4.1. Suppose that jxj < 1 so we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. If in lemma (2.1) we set f(x) = [e.sup.ix] [where [i.sup.2] = -1] we get

[[infinity].summation over (k=0)][e.sup.ik][x.sup.k] = -[[infinity].summation over (m=1)] [i.sup.m-1/m! [[beta].sub.m](x), |x| < 1.

So because [e.sup.ik] = cos k + i sin k we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also we have

[[infinity].summation over (n=0)] [r.sup.n] cos n[theta] = 1 - r cos [theta]/ 1 - 2r cos [theta] + [r.sup.2],

and

[[infinity].summation over (n=0)] [r.sup.n] sin n[theta] = 1 - r sin [theta]/ 1 - 2r cos [theta] + [r.sup.2].

So

[[infinity].summation over (n=1)] [(-1).sup.n][[beta].sub.2n-1](x)/(2n-1)! = 1 - r cos 1/1 - 2x cos 1 + [x.sup.2].

and

[[infinity].summation over (n=1)] [(-1).sup.n][[beta].sub.2n](x)/(2n)! = x sin 1/1 - 2x cos 1 + [x.sup.2].

Therefore proof is complete.

[sections]5. Asymptotic relation between 2-associated stirling number and [B.sub.n](a, b)

We defined the generalized 2-associated stirling numbers by

[[infinity].summation over (n=k)] [S.sup.*.sub.k](n, a, b, k) [t.sup.n]/n! = bt - (1 + t)[a.sup.t]/[a.sup.kt]k!, a [not equal to] 0,

where k and r are positive integers. It is clear that if we set b = e and a = 1 then

[[infinity].summation over (n=k)] [S*.sub.2](n, k) [t.sup.n]/n! = [([e.sup.t] - 1 - t).sup.k]/ k! , see [7].

We give asymptotic expansion of certain sums for generalized 2-associated stirling numbers of the second kind, Bernoulli numbers, Euler numbers by Darboux's method.

Lemma 5.1. Assume that

f(t) = [[infinity].summation over (n=0)] [a.sub.n][t.sup.n]

is an analytic function for |t| < r and with a finite number of algebraic singularities on the circle |t| = r, [[alpha].sub.1], [[alpha].sub.2], ... , [[alpha].sub.l] are singularities of order [omega] is the highest order of all singularities. Then

[a.sub.n] =([n.sup.[omega]-1]/[GAMMA]([omega])(l.summation over (k=1) [g.sub.k]([[alpha].sub.k]) [[alpha].sup.-n.sub.k] + O(r.sup.-n])), see [8]:

Where [GAMMA](w) is the gamma function, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 5.1. Suppose that n [greater than or equal to] 1 and k [greater than or equal to] 1, where k is fixed, when n [greater than or equal to] [infinity], we have (here let ln b/a is a algebraic number)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. It is clear that according to definition we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let

f(t) = [([b.sup.t] - (1 + t)[a.sup.t]).sup.k]/ k![t.sup.k-1]([b.sup.t] - [a.sup.t)[a.sup.kt],

then f(t) is analytic for |t| < 2[pi]/|ln b-ln a| and with two algebraic singularities on the circle |t| = 2[pi]/|ln b-ln a|.

[[alpha].sub.1] = 2[pi]i/|ln b-ln a| and [[alpha].sub.2] = -2[pi]i/|ln b-ln a| are singularities of order 1. To easily we can compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from [8] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore proof is complete.

[sections]6. New method for representation of Apostol Bernoulli and Euler polynomials

Let [sigma](n) denote the set of partitions of n (a nonnegative integer) usually denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [SIGMA] [ik.sub.i] = n

For nonnegative integral vector [bar.k] = ([k.sub.1], [k.sub.2], ... , [k.sub.n]), the multinomial coefficient (x/k) as usual, is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the finite product [PI] runs over i from 1 to n, [(x).sub.k] stands for the all factorial notation, and jkj represents the coordinate sum for the vector [bar.k] = ([k.sub.1], [k.sub.2], ... , [k.sub.n]). Now let

[[g(t)].sup.x] = [summation over (n[greater than or equal to]0)] [A.sub.n](x)[t.sup.n],

where x is an arbitrary complex number independent of t.

Theorem 6.1.([11]) For arbitrary complex number x and y,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So according to these theorems we have the following results.

Corollary 6.1. Let [alpha], fi are complex numbers then we have the following identities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Let [A.sub.n]([alpha]) = [B.sup.([alpha]).sub.n]/n! in theorem 6.1.

References

[1] Qiuming Lue, Euler polynomials of higher order involving the stirling numbers of second kind. (accepted)

[2] Guodong Liu, Generating functions and generalized Euler numbers Proc. Japan Acad. Ser. A Math. Sci. Volume 84, 2(2008), 29-34.

[3] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, cambridge university press, page 147.

[4] C. Frappier, Representation formulas for entire functions of exponential type and generalized Bernoulli polynomials, J. Austral. Math. Soc. Ser., No. 3, 64(1998), 307-316.

[5] Hugh L. Montgomery, Robert C. Vaughan, Multiplicative number theory I. Classical theory, Cambridge tracts in advanced mathematics, 2007, 97.

[6] Boyadzhiev, Khristo N. Apostol-Bernoulli functions, Derivative polynomials and Eulerian polynomials. ArXiv: 0710.1124.

[7] L. Comtet, advanced combinatorics, reidel, 1971 .

[8] M. Wachs and D. White, p, q-Stirling numbers and set partition statistics, j.combin. theory, 56(1990).

[9] D. H. Lehmer, lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals of Mathematics, 1935, 637-649.

[11] S Ramanujan, Some properties of Bernoulli numbers Indian Mathematical Journal, December, 1911.

[11] W. C. Chu and R. k. Raina, Some summation formulate over the set of partitions, Acta Math. Univ. Comenianae Vol. LXI, 1(1992), 95-100.

[12] Hassan Jolany, M. R. Darafsheh, Some relationships between Euler and Bernoulli polynomials. (submitted)

Hassan Jolany and M. R. Darafsheh

School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran
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