Some relations for universal Bernoulli polynomials Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 11. № 4 (2019). С. 130-138.

SOME RELATIONS FOR UNIVERSAL BERNOULLI POLYNOMIALS

M.C. DAGLI

Abstract. In this paper, we consider a generalization of the Bernoulli polynomials, which we call universal Bernoulli polynomials. They are related to the Lazard universal formal group. The corresponding numbers by construction coincide with the universal Bernoulli numbers. They turn out to have an important role in complex cobordism theory. They also obey generalizations of the celebrated Kummer and Clausen-von Staudt congruences.

We derive a formula on the integral of products of higher-order universal Bernoulli polynomials. As an application of this formula, the Laplace transform of periodic universal Bernoulli polynomials is presented. Moreover, we obtain the Fourier series expansion of higher-order universal Bernoulli function.

Keywords: Bernoulli polynomials and numbers, formal group, integrals, Fourier series.

Mathematics Subject Classification 2010: 11B68; 55N22; 42A16

1. Introduction

The higher-order Bernoulli polynomials (x) are usually defined by means of the following generating function:

+ \a ^ +n

^ e" = £ ВП»(,) ^.

7 n=0

For a = 1, they are reduced to classical Bernoulli polynomials Bn(x), The rational numbers Bn = Bn(0) are called classical Bernoulli numbers.

As it is well known, the Bernoulli polynomials play important roles in different areas of mathematics such as number theory, combinatorics, special functions and analysis.

This paper is primarily concerned with a very large class of Bernoulli polynomials. Let us consider the formal logarithm group defined over the polynomial ring Q [c1, c2,...] and the formal power series

s 2 s 3 F (s) = s + C1 - + C2 3 + ....

Let G(t) be its compositional inverse (the formal exponential group):

,2 ,3

G(t) = t- ci- +(3 c2 - 2 02) - + ... 2 6

Some relations for universal Bernoulli polynomials. ©DaCjli M.C. 2019.

This work was supported by The Scientific Research Projects Coordination Unit of Akdeniz University. Project Number: FBA-2018-3593. Submitted January 28, 2019.

The universal higher-order Bernoulli polynomials B^a{x) introduced in [13] and later discussed in [11] and [14] are defined as

if \a ^ tn

(^yj ^ = Bnjx) ^ ^ e R, « = 0. (1.1)

As a = 1 and q = (—1)\ we have F(s) = log (1 + s), G(t) = e* — 1 and the universal higherorder Bernoulli polynomials and numbers reduce to the classical ones. The numbers coincide with the universal Bernoulli numbers defined by Clarke in [4], For brevity, we denote Bca,i(x) = B%(x)md B%a(0) = B^>a.

Because of the generality of definition of universal Bernoulli polynomials, they do not provide many fundamental properties unlike classical case. For example, the higher-order Bernoulli polynomial B^n^ (x) is skew symmetric about x = 2, but there is no symmetry in the universal case, and no obvious root as n > 1 and n is odd.

By construction, for any choice of the sequences {cra}neM they represent a class of Appell polynomials. It means that these polynomials possess the following differential property

d

—5C«W = nB^-i,a(x) as n E N \{0}. (1.2)

The universal Bernoulli numbers provide extensions of the celebrated Kummer and Clausen-von Staudt congruences [1] and general Almkvist-Meurman-type congruences [14]. In [12], it was shown that interesting realizations of polynomials (1,1) can be constructed by means of finite operator theory introduced by G.C, Rota [10], Relations between the universal formal group and the theory of L-series were established in [11, 13].

In this paper, we deduce an explicit formula for the following type integral

X

j K>ai (biz + yi) ■ ■ ■ BC,ar (brz + yr) dz. 0

Since this formula is also valid for the Appell polynomials, the earlier results obtained by Liu,

Pan and Zhang [9], Hu, Kim and Kim [7], Agoh and Dilcher [3] are direct consequences of

the derived formula. As an application of our formula, we present the Laplace transform of

periodic universal Bernoulli polynomials. Furthermore, we obtain the Fourier series of higher-

^ _c

order universal Bernoulli function B (x).

2. Results

2.1. Integral of products of r universal Bernoulli polynomials. In this subsection we describe the integral of products of several universal Bernoulli polynomials.

Theorem 2.1. Let b1,... ,br, y1,... ,yr be arbitrary real numbers obeying bs = 0, 1 ^ s ^ r, and

Ini,..,nr (x; b; y) = /„i,...,„r (x; bi,... ,br; yi,... ,yr)

X

1

if n Bns,*s (h°Z + Vs) dZ, ' 0 -=i

n\ \ • • • n.

Cnu...,nr (x; b; y) = Cnu...,nr (x; bi,...,br ; yi,... ,yr )

1 I r r \

n ^ (b>x + V') - n .

S=i S=i /

nil ••• nr l 1 " * W 11 ns'"s

Then,

Ini,...,nr (x;b;y) = £ (-1)" £ ( . a. )b1 ••• b^b-"-1

, J1 Vr-1ur

n -I I • W 1 1---> Jr— 1

n—O jiH-----hjr-1—" v

• Cn1-il,...,nr-1 -jr-1 ,nr+a+ 1(x; y)

(-1)/+1 (n + /i + 1)!

X /r-1 \ (/+1)

n Bn.s,a.s (b*Z + Vs) ^nr+/+1,ar I yr)

| n ^«s (6 s^ + ?/s) I Bnr+/+1,«r (^ + Vr )

O

where ( i'n ) are the multinomial coefficients defined by

^ ^ n1 + ••• + nr = ^ andn1,... ,nr ^ 0.

( " )

\n1, ••• ,'nrj

yn1} • • • , nrJ n1 ! • • • nr ! In particular, if ^ = n1 + • • • + nr-1, we have

/

n a

J1 ur-1

In i,.,nr (x;b;y) = £ (-1)n £ ( . % V ••• b*;-1

n • i i • n---> Jr— 1

n—O J1+-----hjr-1—a v

Proof. Let

bra lCn1-j1,...,Ur-1-jr-1 ,nr+a+1(X; b; y) • (2-1)

f(z) = B^ai (b 1Z + V1) ■■■ B^r-1,ar-1 (br-1Z + S/r-1) •

We have

X

11 —J f(z)B° tar (brZ + yr)dz =

nr!

br (nr + 1)!^ nr+1'ar 1

( nr + 1)!

Kz)Bt+1,ar Q>rZ + yr)

//(^)Bnr+1,«r (brz + Vr

O

Integrating ^ times by parts, we get:

X/

/n

jf f(z) BZ ,ar (br Z + yr ) dz = £ ( +a+ 1)! [f (a)(z)BZ +a+1,ar (^ + Vr )]

"! O a—0 ( r )!

(-1)

/+1 r

+ (n +1)J ^ (Z)BZr +/+1,«r (^ + yr ) dz.

O

(2.2)

By virtue of the property of a derivative

(fi (z) ■■■ fm (z))(n) = £ ( . ° . W0 ^ ■■■ (z)

— V-?1,..., ^ ms

and (1.2), we arrive at the statement of the theorem. The proof is complete. □

As a consequence of this theorem, we have the following reciprocity relation for Bernoulli polynomials, which generalizes [3, Prop. 2].

X

Corollary 1. For all n,m ^ 0, we have

Y (-1)a(m+ !+a 0 h*h~-a~lBn-aa (b & + yi) BZ+a+^ (b 2X +

a=0 ^ '

- Y (-1)" {ml ! + ^ b*b-a-iB™-aa (^ + V2) Bn+a+i^ (bix + yi) (2.3)

\ lit a /

a=0 v 7

m+n+i

1 , . , m + 1-„ I m + U + 1 \ ,a,TO+ra+i_a dG („.\r>G

+' +i f X- -IX

E (-1)m+i-M 1 +1 M W+ra+i-a^+ra+i-a,7 (yi) fa),

n=0 \ /

hrn+iba+i i 2 a=0

where bi (bi = 0) and yi (1 ^ I ^ r) are real numbers.

Proof. In view of the definition of the integral Ini,..,nr (x; b; y), we see that the left hand side of (2.1) is invariant under interchanging the order of the integrands. We begin by writing the left hand side of (2.1) for r = 2:

t (—otr ;>

a=0 —

{Bt (bix + yi) B<^+a+iIj (b2x + y2) - B%-ari (yi) B<^+a+i fj (y2))

(2.4)

m /

E <-in

n = 0 V

m + n + 1) b«b-«-i , m — a '

a=0

■ {BCm-art (b2X ; y2) BC+a+hl3 (biX ; yi) — B<C-arf (y2) BC+a+hl3 (Vi)) .

Now we can get the reciprocity relation for sums of products of universal Bernoulli polynomials as follows. Let

m ;n ;1\,a,_a_iuC (n.\uC

T := £ (-1)a(m+ 0 b*b-a-iBG^ (yi) BGm+*+^ (V2)

a=0 ^ 7

- it (-1)a(m+ n_ + ^ -a-iB<^-a,1 to) B«+a+Ui (yi) m a

n = 0 V /

This can be rewritten as

c

t=y (-1)n-a(m+^ (y i) Bg+n+i-a!, ( y2)

«=0 ^ 7

ë (-1)m-a (m+r+^ &ra&rm-x7 (î/2) ^i-«,* ( yi).

n = 0 V /

(2.5)

a=0

Without loss of generality we assume that n ^ m; in this case we split the first sum in (2.5)

sum into two sums, one is from 0 to m and the other is from m + 1 to n:

( 1 \n-aj m + ^ + 1 \ un-aua-n-i DG („. \ DG

a=0

(-1)n-a (m+;+^ K-ab2-n-iB^ (y i) B<i+n+l-aIi to)

■+n+i , + I 1 \

Y (-1)m+i-ar +1 brm-X-aBZ+n+i-ar( (yi)B-:, to)

and

(-1)ra-a (m+an+^ K-ab2-n-iBZ( (y i ) ^+ra+i-a;7 (î/2)

E (-i)m+1-a(m +n + ^bnr--1braBC+n+1 -ar/ (y 1) BG, (V2). — m + 1 V /

(—^m+1-a ¡m + n + 1 \ ua-m-1um-a r>G („. \ DG

a—m+1

Hence, we have

t = ^ (-1)m+1-a (m +; + ^ babrn+1-aBZ+n+1-ar( (V1) BZ, (V2). (2.6)

The latter identity and (2.4) give the desired result, i.e., the reciprocity relation for sums of products of universal Bernoulli polynomials. The proof is complete. □

Remark 1. If we start from the left-hand side of (2.3) and argue as in the proof of (2.6), the right hand side of (2.3) turns into

m+n+1 / 1 1\

m + n + 1 \ ,aim+ra+1-a DG /L . „. \ nG

brn+1ba+1 1 2 a—0

This implies that for all x,

m+n+1

+ +1 / + + 1 \

E (-1)m+1-a m + n + ) m+n+1-aBG+n+1-a,1 (b 1X + V1) BG, (b2X + J/2) .

a—0

E (—1)am + n + 1ababm+n+1-aBG+n+1-ar( (b 1X + 1/1) BG, (M +

'+n+1 / + + 1 \

E (-1)a m +an + )babmm+n+1-aBGG+n+1-a,1 (y1)BG,fi

a—0 V /

a—0

m+n+1

Letting 7 = 3 =1, = y2 = 0 in Corollary 1, we arrive at

E (-1)a (m+ 0 ^-^-a (61X) Bm+a+1 (62X) a—0 n - a

- E (-1)a(mm n + ^ bab-a-1BG-a (b2X) BG+a+1 (61X) m a

a—0 N /

i+n+1 ✓ + + 1 \

^ (-1)m+1-a/m+n+M haihm+n+1-aBm a—0 a

m+n+1

= 1 V^ (i)m+1-a/m +n +M wim+n+1-«n G nG

= ^m+1 tfi+1 ( 1) I „ jU1U2 Bm+n+1-aBa

1 2 a—0

Remark 2. A relationship between the reciprocity relation of the Bernoulli polynomials or Euler polynomials and the reciprocity formulae for generalized Dedekind sum Tr (c, d) or Hardy-Berndt sums s3,r (c, d) and s4,r (c, d) was established in [5], [6]. Furthermore, recently, the integral of the product of arbitrary many Euler polynomials was found and by using such formula, several reciprocity relations between the special values of Tornheim's multiple series were obtained by M.S. Kim [8]. In view of this, Corollary 1 may be related to generalized Dedekind sum of the form

sG (c, <0 = t B (n) B G (f),

n—1

_G _

where Br (x) = BG ({x}) and B 1(x) are universal Bernoulli function and ordinary Bernoulli function, respectively. However, this type of Dedekind sum has not been properly defined and considered yet, so it deserves a further study.

_C

2.2. Laplace Transform. In this subsection, we find the Laplace transform of Bn (tu) by means of (2.2).

_c

Let Re s) > 0 and |s/t| < M, where M is constant. By setting f(u) = e-su and using Bn (tu) instead of B^^ (u) in (2.2), we get

X .

W _„.^c,. , , A sat-a-i ( , -=-c

_G ^ ^ g a ^-a-i * _G _G

e Bn (tu) du = (n + a + 1)! 1e Bn+a+i(^x) — Bn+a+i

0 a=0

nlj nK ' ¿-¿(n + a + 1)

_ n — I I v '

d) (n + i + 1)!

(2.7)

(g \ 1 f _Q

t) (n + ¡ + 1)! / ^(t u) du

_c

Since the function Bm (u) = B^ (u — [u]) is bounded, the integrals in (2.7) converge absolutely

_c

and e-SXBc+a+i(tx) tends zero as x ^ <x. Then, letting x ^ <x, we have

'n+a+i

i ,

1 f r tn BG ,(0) Hn+a+i

1 I (+„.\ J„. _ 1 ^n+a+i(0) 5

- e-suBn (tu) du = - — J]

0 a=0

nil n sn+i ^ (n + a + 1)! tn+a+i

a=0 x /

■s\^+i 1 d) (n + i + 1)!

(2.8)

(g \ ^ + i 1 f _Q

t) (n + i + 1)J e-SUB^+i(tu du. 0

Observe that the sum in (2.8) converges absolutely for | s/t| < M as ^ ^ *x>. Also the sequence

of the functions s^B.^ (tu) /¡i\t. converges uniformly in u to zero under the assumption G (t) ^

-¡=tG

e1 — 1. Thus, letting ^ ^ <x and using (1.1), we obtain the Laplace transform of Bc (tu):

o

1 le-BGau)u = r V BC^+i(0) sn+a+i

(n + a + 1)! tn+a+i

0 a=0

tn B°(0) °B°{0)s

(y* B^m i B^m sA

n+i al ta ^ a! ta

\a=0 a=0 J

1s^B%(0)ft Y-a tn s/t

V B°(0)(t\

h a! W

a! \s sn+iG (s/t)'

a=0

2.3. Fourier series. This section is devoted to Fourier series of higher-order universal

_c ,

Bernoulli function Bmr(x). From now on, J denotes the ideal generated by all q — c\ in Q [r] [ci, C2, ..., cm ] .

The higher-order universal Bernoulli polynomials satisfy the following relation [2]

Bm,r(x +1) — Bm,r(x) = —mciB(c-l^-i(x) (mod J). (2.9)

_c

Let us find the Fourier series expansion of Bmr (x). We write

Bm,r (x) ^ ] Cn

n=-rx

The coefficients Cn'^ can be found as

i i

C{:,m) = i Bcr (x)e-2™x = i (x)e

x)= y U^'m)e2mnx.

X

_ , -, Bm+1,r(x)e + ™ i 1 / Bm+1,r(x)e

m + 1 ' O m +17

1 1

2m in I „G , . —2mnx

O

1 (B£+1,r(1) — b"^(0)) +-2^n(nr'm+1)

m + 1 v ^^ ' m+1r^ 'J m + 1

^ -C1BGG,r-1 (0) + m+iCn^ (mod J)

where we have used (2.9) for x = 0. Replacing m by m - 1, we get

C&'m—1) = -C1B<G-lr-1(0) + ^Ct'^ (mod J). - - m

Suppose that n = 0. Then, we get a chain of equivalences mod J:

sy(r,m) — m W r,m-1) , m C1 r>G

Cn " 2mnCn + 2rnnBm-1'r-1(0)

m (m - 1 C(r,m-2) , (m - 1) C1 BG (0)A + mc1 BG (0)

to Vl^TCn ) + 2^n m—2,r+ ~2ij/flBm—^'r-1(0)

2m z n

m (m - 1) c(r,m-2) + m (m - 1) C1 BG (0) + BG (0)

— /ri . -,2 Cn + /ri . s 2 nm-2,r-1(0) + o • Bm-1,r-1(0)

(2mn)2 (2mm)2 ' 2mm '

_ m (m — 1) fm—2 (r,m-3) (m — 2) C^g

— • N2 o • Cn + o • Bm-3,r-1(0)

(2mn)2 \ 2mn 2mm ' J

m ( m - 1) 1 G m 1 G

+--• N2-Bm-2,r-1(0) + ^—Bm,-1,r-1(0)

(2mn)2 ' 2mn '

= m (m — 1) (m - 2) c(r,m-3) + m (m — 1) (m - 2) C1 BG (0)

— in ■ \3 Cn + . ,3 Bm-3,r-1(0)

(2m n) (2mn)

m ( m - 1) 1 G m 1 G

+--• N2-Bm-2,r-1(0) + Bm-1,r-1(0)

(2mn)2 ' 2mn '

Proceeding as above, we obtain that

C{nr,m) — m (m —1)(mm-12) ••• 2 cnr1 + C1 V -tmkrB* -1(0) (mod J) , (2.10)

m - 1

__'™>k dg

Zo • \m-1 cn + ^ . sk

(2m n) k—1 (2mm)

where (x) n = x (x — 1) ••• (x — n + 1) Using the identity BGr (x) = — d ^x — - j given in [2], we can compute the coefficients Cn'1 in the last identity as follows:

Cir'1) = f BGr (x)e-2mnx = — f d(x —r-) e-2mnx- C1

2) 2m n

OO

Substituting this identity in (2.10) gives

m 1

c(r,m) = m!C1 +r sr^ (m)k BG (0)

cn — /n -\m + C1 / . . skBm-k,r-1(0)

(2mn) (2mn)k '

m

— ^E 7!rhVkB^-k,r-1(0) (mod J) k—1 (2mn)

As n = 0, we readily see that

i

c0r'm) = / Kr(*) = K+i,(1) - ^(0))

0

yG

= -CiBrntr-i(x) (mod J).

This implies:

= 0.(r .m^ina

^^ (x) = J] C*

n=-i

= -dBg,-i(x) + a Y (E l^mh^-Kr-I(0)) e27Tinx

=-i\k=i (2-n) J

™ (m) ( l p2mnx \

aBg^i(x) - c £ ^Bg^_1(0) -H! E (j-r-T feÎ V ni-i (2-m) /

™ / \

= - CiBZir-i (x) - ciYl%)BZ-k,r-i(0)Bk (x)

k=2 ^ '

^ ,.=-i (2- ^ \nG . J0YR,.(x)

- -i(0^0 ifx e z,

m _

-CiE{™)Bg-k,r-i(0)Bk (x) f e Z;

k=0

" (m\r,G ^ (modJ)

-ciE(™)Bg-k,-i(0)Bk(x) ifx e Z,

k=0 k=i

where we have used the Fourier series expansion of ordinary Bernoulli function Bm(x)

—— , , m! ^ t e27Tinx

Bm(x) = — --^ > -, m ^ 2,

my ' (2-i)m ^ nm ' '

n=-l n=0

and the fact

" e2lTinx _ \Bi(x) ifx e Z; 2-in = |0 ifx G Z.

l „2%inx I td i

_ v -_= \Bi(

^ 2-i n I 0

n=-l V n=0

Thus, we have proved the following theorem.

_Q

Theorem 2.2. For m,r ^ 2 and x e (-œ, +œ) , Bmr(x) has the Fourier series expansion BI,(x) =-ClBlr-i(x) + Cl jt (it 7TT^^r-i(0) ) e27Tinx (mod J).

ra=-i Vk=i (2™0 )

n=0

Moreover, for x e (-œ, +œ)

m / \

£(m)

k=1

_Q __^ / ^^ \ _

B(x) = -Ci > )BZ-k,r-i(0)Bk(X) (mod J) ,

where Bk (x) is the ordinary Bernoulli function.

REFERENCES

1. A. Adelberg. Universal higher order Bernoulli numbers and Kummer and related congruences // J. Number Theory. 84:1, 119-135 (2000).

2. A. Adelberg. Universal Bernoulli polynomials and p-adic congruences, applications of Fibonacci numbers //in "Proceedings of the Tenth International research conference on Fibonacci numbers and their applications". Springer, Dordrecht. 9, 1-8, (2004).

3. T. Agoh, K. Dilcher. Integrals of products of Bernoulli polynomials //J. Math. Anal. Appl. 381:1, 10-16 (2011).

4. F. Clarke. The universal von Staudt theorems // Trans. Amer. Math. Soc. 315:2, 591-603 (1989).

5. M.C. Dagli, M. Can. On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials //J. Number Theory 156, 105-124 (2015).

6. M.C. Dagli, M. Can. On the integral of products of higher-order Bernoulli and Euler polynomials // Funct. Approx. Comment. Math. 57:1, 7-20 (2017).

7. S. Hu, D. Kim, M.S. Kim. On the integral of the product of four and more Bernoulli polynomials // Ramanujan J. 33:2, 281-293 (2014).

8. M.S. Kim. On the special values of Tornheim's multiple series //J. Appl. Math. & Inform. 33:3-4, 305-315 (2015).

9. J. Liu, H. Pan, Y. Zhang. On the integral of the product of the Appell polynomials // Integ. Transf. Spec. Funct. 25:9, 680-685 (2014).

10. G.C. Rota. Finite operator calculus. Academic Press, New York (1975).

11. P. Tempesta. L-series and Hurwitz zeta functions associated with the universal formal group. Annali Scuola Normale Superiore, Classe di Scienze. 9:5, 133-144 (2010).

12. P. Tempesta. On Appell sequences of polynomials of Bernoulli and Euler type //J. Math. Anal. Appl. 341:2, 1295-1310 (2007).

13. P. Tempesta. Formal groups, Bernoulli-type polynomials and L-series // C. R. Math. Acad. Sci. Paris, Ser. I 345:6, 303-306 (2007).

14. P. Tempesta. The Lazard formal group, universal congruences and special values of zeta functions // Trans. Amer. Math. Soc. 367:10, 7015-7028 (2015).

Muhammet Cihat Dagli, Department of Mathematics, Akdeniz University, Dumlupinar Boulevard, 07058 Campus, Antalya, Turkey

E-mail: [email protected]