Generalized Bernoulli numbers and polynomials in the context of the clifford analysis Текст научной статьи по специальности «Математика»

УДК 517.55+517.962.26

Generalized Bernoulli Numbers and Polynomials in the Context of the Clifford Analysis

Sreelatha Chandragiri*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Institute of Science, Technology and Advanced Studies

Vels University Pallavaram, Chennai, 600117 India

Olga A. Shishkina^

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 31.05.2017, received in revised form 10.08.2017, accepted 20.11.2017 In this paper, we consider the generalization of the Bernoulli numbers and polynomials for the case of the hypercomplex variables. Multidimensional analogs of the main properties of classic polynomials are proved.

Keywords: hypercomplex Bernoulli polynomials, generating functions, Clifford analysis. DOI: 10.17516/1997-1397-2018-11-2-127-136.

Introduction

The Bernoulli polynomials for natural values of the argument were first considered by J. Bernoulli (1713) in relation to the problem of summation of powers of consecutive natural numbers. L.Euler studied such polynomials for arbitrary values of the argument, the term "Bernoulli polynomials" was introduced by J.L.Raabe (1851).

The Bernoulli numbers and polynomials are well studied and find applications in fields of pure and applied mathematics. Various variants of generalization of the Bernoulli numbers and polynomials can be found in [5-11]. A generalization to several variables has been considered in [12]; in this paper definitions of the Bernoulli numbers and polynomials associated with rational lattice cones were given and multidimensional analogs of their basic properties were proved.

This paper is devoted to generalization of these results to the case of hypercomplex variables. The Clifford algebra in hypercomplex function theory (HFT) was first used by R. Fueter [1] in the beginning of the last century. A systematic study of this topic can be found in [2-4]. Also, the papers [15-18] with further advancement of the Clifford analysis should be noted. The notion of the Bernoulli numbers and polynomials in this framework were given and studied in [13,14]. In this paper we give a more genral notion of Bernoulli polynomials than in [13,14], namely, in the spirit of [12] we define polynomials in hypercomplex variables associated with a matrix of integers. In the second section of the paper we formulate and prove basic properties of such polynomials.

* [email protected] t [email protected] © Siberian Federal University. All rights reserved

1. Notation and definitions of the generalized Bernoulli polynomials and Bernoulli numbers

Let {e1,... ,en} be an orthonormal base of the Euclidean vector space Rn with a product according to the multiplication rules

ekei + eiek = —2Skh k,l = 1,...,n,

where 5kl is the Kronecker symbol. This non commutative product generates the 2n-dimensional Clifford algebra Clo,n over R and the set(eA : A C {1,..., n}) with eA = ehl eh2 .. .ehr, 1 < h1 < ... ^ hn, e$ = eo = 1, forms a basis of Clo,n. The real vector space Rn+1 is embedded in Clo,n by identifying the element (xo, x1,..., xn) G Rn+1 with

2 = xoeo + X1e1 +----+ Xnen G A = spanR {eo,..., en} = Rn+1.

The natural generalization of the complex Cauchy-Riemann operator

d = 1( d dz 2 \ dx dy J

is given by the operator

and the equation

d d d

D = ---+ -— e1 + ... + -— en

oxo 0x1 oxn

Df = 0

defines hypercomplex holomorphic (or monogenic ) functions f = f (z) as Clifford algebra valued functions in the kernel of this generalized Cauchy-Riemann operator (cf. [15]). Since the operator D can be applied both from the left and from the right hand side of f, it is usual to refer to it as a left monogenic function or a right monogenic function, respectively. For simplicity, from now on we only deal with left monogenic functions. The case of right monogenic functions can be treated completely analogously.

Since Dz = 1 — n it is evident that the function f (z) = z G A is only monogenic if n = 1, i.e., in the case of A = C. This implies significant differences between the cases n =1 and n > 1. Moreover, powers of z, i.e., f (z) = zk, k = 2,..., are not monogenic which means that they cannot be considered appropriate as hypercomplex generalizations of the complex power zk, z G C. These facts are the reason for generalized power series of a special structure, which we are going to use in the following subsection.

To overcome the mentioned situation, in [16] has been considered another hypercomplex structure for Rn+1 and

Hn = {z : z = (z1,..., zn), zk = xo — xk ek, xo, xk G R k =1,.. .,n},

whereas the components of the vector z, i.e. the hypercomplex variables zk themselves are monogenic, their ordinary products zizk, i = k, are not monogenic. But a n-ary operation, namely their permutational (symmetric) product resolves the problem (cf. [16]).

Definition 1. Let V+ be a commutative or non-commutative ring, ak G V (k = 1,... ,n), then the symmetric "x " product is defined by

a1 x a2 x • • • x an = ^ Y^ ahai2 ••• an, (1)

n! z—'

n(il,...,in)

where the sum runs over all permutations of all (i1,... ,in).

Additionally, the following convention has been introduced in [16]. Convention. If the factor aj occurs aj times in (1), we briefly write

a1 x • • • x ai x • • • x an x • • • x an = a^1 x • • • x = a7, (2)

^ S/ ^ ** V **

where a = (a1,... ,an) £ Nn and set parentheses if the powers are understood in the ordinary way.

Formula (2) simply allows to work with a polynomial formula exactly in the same way as in the case of several commutative variables. It holds

(Z1 + ■■■ + zn)k = Z (k)zi1 x ■■■ x n = Z ( ka)ga, k G N (3)

n) = z^ \ a Z1 ^ \ a

= k V У \v\ = k

with polynomial coefficients defined as usual by ^ = _!(1"" , where a! = a1! + ■ ■ ■ + an!

k \ =_k[_

a J = a!(k - a)!' (see [17,18]).

In [17] it has been shown that the partial derivatives of zu with respect to xk are obtained as

dza

— = akz°-Tk, (4)

dxk

where rk is the multiindex with 1 at place k and zero otherwise.

It is well known that for complex holomorphic functions f : C ^ C the complex derivative df

f' = — exists and coincides with the complex partial derivative

dz

df =li df _ f

dz 2 \ dx dy J

The analogous situation is true in the hypercomplex case (cf. [18]). A real differentiable function f (z) is left (right) hypercomplex derivable in Q с Hn if and only if f is left (right)monogenic in Q С Hn. In the case of its existence, the hypercomplex derivative is given by

2Df resp.1 fD,

with the conjugated generalized Cauchy-Riemann operator

-¡^ д д d

D = ä--ä— ei----- -— en.

dxo dxi dxn

Furthermore, like in the complex case, where the complex derivative satisfies

f, = df = df

dz dx'

the left(right) hypercomplex derivative of f at z is exactly

2 Df =2 fD = f

2 2 dx0

Let a1,.. .,an be vectors with real coordinates aj = (aj,..., ajn) and

A-

( a\ ...... a\\

\ an ...... an

is a matrix with coordinates of the vectors aj in a column.

l

an

a

2

2

Definition 2. The hypercomplex Bernoulli polynomials BM (z) = ,..., ..., zn),

G к = 1,... ,n associated with the matrix A are defined as the coefficients of a multiple power series ordered with respect to the degree of homogenity by the following relation

<t,z>

n / œ j k\ œ

П £ Щ-) £=

j = l \ k=0 v ' J |ß| = 0

tß

bm j,

(5)

where <t,z > = t\z\ + ... + tnZn, z = (zi,..., zn), tß = tß1 ... tßn, jl = ¡л\\... jnl,

\ J \ = ji + ..., +Jn-

Applying (3), the formula (5) is equivalent to

T-

^ al

z X • • • X zn

H=0

œ i œ tß

t' = E (¡rW»'()" 5= Bß(z) j:

|"| = 0 Vl 1 ' |ß|=0 ^

where s = (s1,...,sn), a = (a1,...,an) and (ajt)" = (a{t1) 1.... (ajntn)"n.

Comparing both sides gives the relationship of hypercomplex Bernoulli polynomials to the generalized powers

J2 J2 П(\ sj \ +l)sj l)

a+ß=a "1 + ... + " n=a\j=1 "•il.'/

,it)"1 (ant)"n Bß(z) = 1 jl al

(a1t) . . . (ant)

- ziff1 x z2ff2 x • • • x znan, (6)

where a = (ai,...,an), J = (ji,...,jn), t = (ti,...,tn), z = (zu...,zn), a = (ai,...,an), sj = (s{,...,sjn) for ak =0,1,... (к = 1,..., n).

Obviously, the set of hypercomplex Bernoulli polynomials contains n copies of the classical

( 1 .. 1 1

Bernoulli polynomials that are obtained when matrix A = ......... and all the indices

V 1 ... 1 )

jk, к = 1,... ,n in (6) are equal to zero or only one of them is different from zero.

For example, some hypercomplex Bernoulli polynomials given by (6), for n = 2, are equal to

B0,0(z Bifi(z

B0,i(z

Bi,i(z

B2,0(z

zi - ö (ai + a2)

z2- ö (a2 + a2)

zi x z2 - 2

zi 2\ / 1 2\ /12 1 2\ /11 2 2\

ziy a2 + a2) + z2{ a1 + aj + - {a1a2 + a2 a^ + - {a1a2 + a1a2)

z2 - z^al + a1) + 1 ((a1)2 + (a2x)^ +

11 a21 .

Definition 3. Generalized Bernoulli numbers Bai. an are the values of the Bernoulli polynomials at the origin

Bai,...,an = Bai,...,an (0, . . . , 0).

1

1

1

2

For instance, for n = 2 we obtain next values of Bernoulli numbers: Bo,o =Bo,o(0,0) = 1,

Bi,o =Bi,o(0,0) = -1 (ai + aj),

1

Bo, l =Bo,i(0,0) = - 2 (a? + a?)

^a? + alal) + 6 a-1 ' B?,o =Bi,o(0, 0) = U(a 1 )? + (a?)1) + 1 a 1-1.

Bi 1 =Bi i(0, 0) = - (ala? + a?a?) + - (alal + ala?),

4 4 6

2. Properties of Bernoulli numbers and Bernoulli polynomials

Property 1. Hypercomplex Bernoulli polynomials and generalized Bernoulli numbers satisfy the expression

Bau...,an (I,---, 1) = (-1) 1 ]Bai (7)

Proof. Making use of the definition of hypercomplex Bernoulli polynomials by generating function

œ -

F (£*)=£ - Bai, ...,an (Zl,...,zn) t? ...tnn ,

I a I =0 '

where

< a ,t > e<z,t> e<aj ,t> _ - '

F (t,z) = П j=l

and taking (z1;..., zn) = (0,..., 0) and (z1;..., zn) = (1,..., 1) we get

œ 1

F (t 0)= Z - Bai,..,an t?1.....tn

z—' — !

I a I =0

and

œ 1

F (t, 1)= E 1 Bai...,an (1,..., 1) ta1.....tnn ,

-!

a =o

respectively.

Moreover F(t, 1) = F(—t,0), that is,

œœ

E "Г Bai...,an (1,..., 1) t?1 ...tnn = E I Bai,...,an Hl)? ... Hn)^ = -! -!

a =o a =o

œ

= £ "Г Bai,..a (-1) 1 ? 1 tai ...tnn . -!

a =o

Hence,

Bai,...,an (1,..., 1) = (-1) 1 ? 1 B,

ai.....an-

□

The equality (7) generalizes the property Bn(1) = ( — 1)nBn, n £ N0, already known in the classical case.

Property 2. Hypercomplex Bernoulli polynomials can be expressed by generalized Bernoulli numbers as

Bai,...,an (zl,...,zn) = £ •••£ ( 1 ) •••[ f) Bji.j zai-ji X •• X znn-jn. (8)

ji = 0 jn=0 ^ ' n

Proof. Using the definitions of hypercomplex Bernoulli polynomials and the Bernoulli numbers, we can write

E( ST B ^ X • • • X zn \tai ta n = œ Bai (zl, ...,Zn) tai a

Д .+-* Bji'-'jn jl! ••• jnlAl! ••• kn! tl •••tn = .¿=0 -l! ...-n! tl •••tn

I a I =0 j + k=a ' I a | =0

which yields

Bai,...,an (z1,...,zn) = 52 "^52 Bji

that is,

a a B -l! ...an!za1i-ji X •••x zQn-jn

<7i'"" <7nV~i"' "Zn) i=0 ••• l=0 Bji'"j 1l! ...jn!(-l - jl)!... — - jn)!,

B(zi,...,zn)= Y, •••Yy j •••[ jn Bji'-jn z^1 x ••• x zln-3n. □

ji = 0 jn=o ^ ' \ n /

With (8) we found a generalization for another property of the classical Bernoulli polynomials

Bn(z) : n

Bn(z) = ПЬ zn-k, n G No.

k=0 ^ '

Proposition 2 still allows to introduce a new type of Bernoulli numbers, where one of the arguments is equal to one and the others are equal to zero, which is a situation different from that one in Proposition 1, which describes the symmetry relation between Bai, ... ,(1,..., 1) and

Bai,... ,an ■

Property 3. Let us call k-Bernoulli numbers, B1k1... Œn, those that are obtained by calculating the hypercomplex Bernoulli polynomials in (0,..., 1 ,..., 0), k = 1,...,n, i.e.,

Bki, ...,*n = Bai, „„ (0,..., ^ ,..., 0).

k

Then these k-Bernoulli numbers can be represented as linear combinations of the generalized Bernoulli numbers,

E( Bai,j.... о

Bki,...,an = V ( .k )B„

jk = 0

Proof. The proof follows immediately from (8) by taking zk = 1 and zi = 0, i = 1,... ,n, i = k. □ Example.

Bji = B2 ,i(1,0) = 1Bo ,i + 2Bi ,i + 1B2 ,i

B3, 2 = B3,2 (1, 0) = 1Bo,2 + 3Bi,2 + 3B2,2 + 1B3,2

B\ 3 = B4 3 (1,0) = 1Bo,3 + 4Bi,3 + 6B2,3 + 4B3 3 + 1B4 3

B2,1 = B11 (0,1) = 1B1,o + IB11

B2ia = Bh2(0,1) = 1B1,o + 2B11 + IB12

B|,3 = B2,3(0,1) = 1B2,o + 3B2,1 + 3B22 + 1B23

Property 4. We have

d^B^n (z1, ...,zn) = \ (z1, ...,zn) ,«k = 0 k =!,... n

dxk ' ' [0 ,ak = 0,

where zk = xk — xoek, xo, xk G R, k = 1,..., n

Proof. The proof follows directly by partial differentiation with respect to xk of both sides of (8) together with (4). □

This proposition generalizes for the hypercomplex case the relations B'n(z) = nBn-1(z), n G N, used for the differentiation of classical Bernoulli polynomials.

Property 5. For the hypercomplex derivative of Ba(z) of ^DBa(z) the next property holds true 1— n

2 DBai,...,an (zU ...,zn) = — ak B*l,...,Vk(z1, zn)ek .

k=1

Proof. Considering that the hypercomplex Bernoulli polynomials are monogenic, i.e.,

DB„i,...,an (z1, ...,zn)=0,

we can write

d n d — B«i,...,«n (z1, ...,zn) = — £ dxk B'i-."-'n (z1, . ., zn)ek , o k = 1 k

that is

1 —

- DBai.,,an (zi, ...,zn) = - au Bau..,ak(zi,..., zn)ek.

k=i

□

Let S = (S1;..., Sn), Sj is a shift operator Sj f (z) = f (z1,... ,zj + 1,..., zn).

To formulate the next property we have to introduce some notation. Denote by Q(S) =

n - 3 3 n

П (Sa — 1), the linear operator where Sa = S/ ..., Snn. Denote by daj = < aj ,d > = j dk j=i k=i the differential operator along the vector aj, and let da = dai ... dan.

Property 6. Hypercomplex Bernoulli polynomials satisfy the equation

Q(S)Bau..,an (zi, ...,zn ) = dazl1 X---X zln . Proof. Note that (Sa3 — 1)e<g'r> = (e<a3>*>, -1)в<*'Г>, then

nn

J](Sa3 — 1)e<*'r> = ]J(e<a3 ¿> — 1)e<z'r>. j=i j=i

It means that operator Q(S) acts on e<z't> by the formula

n

Q(S)e<z'z> = Y[(e<a3't> - l)e<z'z>. (9)

j=i

Now we will show how the operator Q(S) acts on the generating function F(t, z) of Bernoulli polynomials.

n j t Q(*)Ftf z) = Il j^lQ(S)e<z'Z>. j=i

Using (9) we obtain

n j t n QWF (?,*) = n U(e<ai 'Z> - l )e<z'Z> ■

j=i e 1 j=i

As a result we have

n

Q(S)F(t,z) = H < aj,t >e<z'Z>. j=i

Since da. e<z'z> = < aj ,t >e<z'z>, then

3ae<Z'Z> = JJ < aj,t >e<z'z> j=i

Using the definition of Bernoulli polynomials and acting by operator Q(S) on each part, we obtain

Q(s)njV** = Q(S) ± ;Zn) r ■

j = 1 M = 0 x"" J

Since Q(S)F(t,z) = dae<z'z>, then

<Z'Z> _ _ ST Q(S) B°l'-'°n (Zl,...,Zn) t<T

dae<Z'z> = ]T

ail ...aj.

CO

<a z? x---x zj- ^ Q(S) Bau..„an(z1?...,zn) t°

Eda Z1 X ■ ■ ■ X zn ta = V^

ail ...aj. t =

M=0 1 |ct|=0

ail ...anl ail ...aj.

Q(S)Bru... (zi,...,zj) = dazl1 X ■■■x zjn

This work of first author was financed by the PhD SibFU grant for support of scientific research no. 14.

This work of second author is supported by the Russian Federation Government grant to conduct research under the guidance of leading scientists at Siberian Federal University (contract 14.y26.31.0006) and was financed by the grant of the President of the Russian Federation for state support of leading scientific schools NSh-9149.2016.1.

The reported study was funded by RFBR according to the research project no. 18-31-00232.

References

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[2] R.Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann., 185(1970), 91-111.

[3] K.Gurlebeck, K.Habetha, W.Sprobig, Holomorphic functions in the plane and n-dimensional space, Birkhauser Verlag, Basel, 2008.

[4] H.R.Malonek, M.I.Falcao, On paravector valued homogeneous monogenic polynomials with binomial expansion, Advances in Applied Clifford Algebras, 22(2012), no. 3, Sl, 789-801.

[5] G.Bretti, P.Natalini, P.E.Ricci, Generalizations of the Bernoulli and Appell Polynomials, Abstr. Appl. Anal, (2004), 613-623.

[6] F.Costabile, F.Dell'Accio, M.I.Gualtieri, A new approach to Bernoulli Polynomials, Rendi-conti di Mathematica, 26(2006), Series VII, Roma, 1-12.

[7] V.Lakshmikantham, D.Trigiante, Theory of Difference equations, Numerical Methods and Applications, M.Dekker, New York, 2002.

[8] P.Natalini, A.Bernardini, A generalization of the Bernoulli Polynomials, J. Appl. Math., (2003), 155-163.

[9] Z.Zhang, J.Wang, Bernoulli Matrix and its algebraic properties, Discrete Appl.Math., 154(2006), no. 11, 1622-1632.

10] Y.Simsek, Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta functions and L-function, J. Math Anal. Appl., 324(2006), no. 2, 790-804.

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Обобщенные числа и полиномы Бернулли в контексте клиффордова анализа

Шрилата Чандрагири

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Институт науки, технологии и современных исследований

Уиверситет Велса Паллаверан, Ченнаи, 600117 Индия

Ольга А. Шишкина

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

В 'работе 'рассматривается обобщение чисел и многочленов Бернулли для случая гиперкомплексных переменных. Доказаны многомерные аналоги основных свойств классических чисел и многочленов Бернулли.

Ключевые слова: гиперкомплексные многочлены Бернулли, производящие функции, клиффордов анализ.