R E S E A R C H
Open Access
Differential equations for the extended 2D
Bernoulli and Euler polynomials
Banu Yılmaz
*and Mehmet Ali Özarslan
**Correspondence:
[email protected]; [email protected] Eastern Mediterranean University, Mersin 10, Gazimagusa, TRNC, Turkey
Abstract
In this paper, we introduce the extended 2DBernoulli polynomials by
tα (et– 1)αc
xt+ytj =
∞
n=0
B(α,n j)(x,y,c)t
n
n!
and the extended 2DEuler polynomials by
2α (et+ 1)αc
xt+ytj= ∞
n=0
E(α,n j)(x,y,c)t
n
n!,
wherec> 1. By using the concepts of the monomiality principle and factorization method, we obtain the differential, integro-differential and partial differential equations for these polynomials. Note that the above mentioned differential equations for the extended 2DBernoulli polynomials reduce to the results obtained in (Bretti and Ricci in Taiwanese J. Math. 8(3): 415–428, 2004), in the special casec=e,
α
= 1. On the other hand, all the results for the second family are believed to be new, even in the casec=e,α
= 1. Finally, we give some open problems related with the extensions of the above mentioned polynomials.MSC: Primary 11B68; secondary 33C05
Keywords: 2DBernoulli polynomials; 2DEuler polynomials; 2DAppell Polynomials; Hermite-Kampé de Fériet (or Gould-Hopper) polynomials; differential equations; generalized heat equation
1 Introduction
A polynomial set{Pn(x)}∞n=is quasi-monomial if and only if there exist two operatorsPˆ
andM, independent ofˆ n, such that
ˆ
PPn(x)
=nPn–(x) and Mˆ
Pn(x)
=Pn+(x).
Here,Mˆ andPˆplay the role of multiplicative and derivative operators, respectively. Owing to the fact that every polynomial set is quasi-monomial [], by using the monomiality prin-ciple, new results were obtained for Hermite, Laguerre, Legendre and Appell polynomials in [–].
In this paper, we consider Appell polynomials. Before proceeding, we recall some basic definitions and properties of the polynomial families that we discuss throughout the paper.
The celebrated Appell polynomials can be defined by the following generating relation:
GA(x,t) =A(t)ext=
∞
n=
Rn(x)
tn
n!, ()
where
A(t) = ∞
k=
Rk
tk
k!, A()=
is an analytic function att= andRk:=Rk(). Assuming that
A(t) A(t) =
∞
n=
αn
tn
n!
it is easy to see that for anyA(t) the derivatives ofRn(x) satisfy
Rn(x) =nRn–(x).
Lettingn:=nDx, whereDx:=dxd, it is straightforward that
(· · ·n–n)Rn(x) =R(x).
On the other hand, it is shown in [] that, if
n:= (x+α) +
n–
k=
αn–k
(n–k)!D
n–k x
thenn(Rn(x)) =Rn+(x). Hence, we have the following relation:
(n+n)Rn(x) =Rn(x). ()
Sincenandnare differential realizations, equation () gives the differential equation
that is satisfied by Appell polynomials []. In [], M.X. He and Paolo E. Ricci obtained the differential equations of the Appell polynomials via the factorization method. Moreover, they found differential equations satisfied by Bernoulli and Euler polynomials as a spe-cial case. Afterward, Da-Qian Lu found differential equations for generalized Bernoulli polynomials in [].
Bernoulli polynomials are defined by the following generating relation:
G(x,t) = t et– e
xt=
∞
n=
Bn(x)
tn
n!; |t|< π
and Bernoulli numbersBn:=Bn() can be obtained by the generating relation
t et– =
∞
n=
Bn
tn
Particularly,
B= , B=
–
, B=
()
andBk+= for (k= , , . . .). Bernoulli numbers play an important role in many
mathe-matical formulas. For instance,
• MacLaurin expansion of the trigonometric and hyperbolic tangent and cotangent functions,
• the sums of powers of natural numbers,
• the residual term of the Euler-Maclaurin quadrature formula [].
Bernoulli polynomials, first studied by Euler [], are employed in the integral representa-tion of differentiable periodic funcrepresenta-tions, and play an important role in the approximarepresenta-tion of such functions by means of polynomials [].
First, the three Bernoulli polynomials are
B(x) = , B(x) =x–
, B(x) =x
–x+
. ()
The following properties are straightforward:
Bn() =Bn() =Bn, n= ,
Bn(x) = n
k=
n k
Bkxn–k,
Bn(x) =nBn–(x).
TakingA(t) =
et+in (), we meet with the well-known Euler polynomials. More precisely,
the Euler polynomials are defined via the generating relation
GE(x,t) =
ext
et+ =
∞
n=
En(x)
tn
n!, |t|<π.
On the other hand, the Euler numbersEnare defined by the following relation:
et+e–t =
∞
n=
En
tn
n!.
Moreover,
En
= –nEn
and (see in [, ])
ek= –
k
k
h=
k h
Ek–h.
Recently, Gabriella Bretti and Paolo E. Ricci defined the two-dimensional Bernoulli
poly-The two-dimensional Euler polynomialsE(nj)(x,y) are defined as
G(j)(x,y;t) =
They obtained explicit forms of the polynomialsB(nj)(x,y) by means of Hermite-Kampé de
Fériet (or Gould-Hopper) polynomialsHn(j)(x,y), where these polynomials are defined by
ext+ytj=
Furthermore, Gabriella Bretti and Paolo E. Ricci gave a recurrence relation, shift operators, differential, integro-differential and partial differential equations for two-dimensional Bernoulli polynomials in []. We gather these results in the following theorem:
Theorem [] Gabriella Bretti and Paolo E. Ricci For n∈N,the recurrence relation of theD Bernoulli polynomials is given by
B(j)(x,y) = ,
Shift operators are given by
L–n:=
Differential,integro-differential and partial differential equations are
x–
Dy+jD–(j–)
x Djy–+jyD–(j–)
x Djy
–
n–
k=
Bn–k+
(n–k+ )!D
–(j–)(n–k)
x Dny–k+– (n+ )Djx–
B(nj)(x,y) = , ()
x–
Dx(j–)(n–)Dy+ (j– )(n– )Dx(j–)(n–)–Dy+jD(xj–)(n–j)
Djy–+yDjy
–
n–
k=
Bn–k+
(n–k+ )!D
(j–)(k–)
x Dny–k+– (n+ )D(xj–)n
B(nj)(x,y) = , n≥j ()
respectively.
From here and throughout the paper,
Dx:=
d
dx, Dy:= d
dy, D
–
x f(x) := x
f(ξ)dξ.
Note that Gabriella Bretti and Paolo E. Ricci investigated the casej= separately. In this paper, we consider the D extension of the Bernoulli and Euler polynomials. To obtain the explicit forms of them, we take into consideration of the extended Kampé de Fériet (or Gould-Hopper) polynomials. Let us define the extended Hermite-Kampé de Fériet (or Gould-Hopper) polynomials by the following generating relation:
cxt+ytj= ∞
n=
P(nj,c)(x,y)t
n
n!, c> . ()
It is clear thatPn(j,c)(x,y) is explicitly given by
P(j,c)
n (x,y) =n!
[nj]
s=
xn–jsys
(n–js)!s!(lnc)
n+s–js, ()
wherej≥ is an integer. Note thatc=e, givesPn(j,c)(x,y) =Hn(j)(x,y) where
Hn(j)(x,y) =n!
[nj]
s=
xn–jsys
(n–js)!s!
are Hermite-Kampé de Fériet (or Gould-Hopper) polynomials.
It is meaningful to mention that the polynomialsPn(j,c)(x,y) are very important in solving
the generalized heat equation:
(lnc)–j ∂
j
∂xjF(x,y,c) = ∂
∂yF(x,y,c),
Moreover, other generalizations which includeP(nj,c)(x,y) polynomials can be defined by
Gabriella Bretti and Paolo E. Ricci gave the explicit form of D Bernoulli polynomials by
B(nα,j)(x,y) =
On the other hand, generalized Bernoulli and Euler polynomials were defined by H. M. Srivastava, Mridula Garg and Sangeeta Choudhary in [] as follows.
Leta,b,c∈R+(a=b) andn∈N
. Then the generalized Bernoulli polynomialsB(
α)
n (x;λ;
a;b;c) of orderα∈Care defined by the following generating relation:
of orderα∈Care defined by the following generating relation:
These definitions motivate us to consider the following extended DBernoulli and Euler polynomials:
Definition The extended DBernoulli polynomials of orderαis defined as
tα
ized DBernoulli polynomials. Note that the generalized Bernoulli numbers are defined by
Definition The extended DEuler polynomials of orderαis defined as
Note that in the casec=ein (), we call the polynomialsEn(α,j)(x,y) :=E(
α,j)
n (x,y,e), as the
generalized DEuler polynomials.
In the following section, we obtain the explicit forms of the Dextension of Bernoulli polynomials, by means of Hermite-Kampé de Fériet (or Gould Hopper) polynomials and Bernoulli numbers. Moreover, we obtain differential, integro-differential, partial differ-ential equations and shift operators for the extended DBernoulli polynomials by us-ing the factorization method, introduced in []. We list the results for the extended D Bernoulli polynomials. In Section , we deal with finding the recurrence relation, differen-tial, integro-differential and partial differential equations for the extended DEuler poly-nomials. Finally, in Section , we present some open problems that will be investigated in the future.
2 2Dextension of generalized Bernoulli polynomials and their differential equations
We begin by the following theorem that gives the explicit form of extended DBernoulli polynomials via Hermite-Kampé de Fériet (or Gould Hopper) polynomials:
Theorem The relationship between P(nj,c)(x,y)and B(
α,j)
n (x,y,c)is given by
B(α,j)
n (x,y,c) = n
k=
n k
Pk(j,c)(x,y)Bα
n–k, c> , ()
where Bkdenotes the Bernoulli numbers.
Proof Since
∞
n=
B(nα,j)(x,y,c)t
n
n! = tα (et– )αc
xt+ytj
the result is obtained by using () and () and then applying the Cauchy product of the
series.
Corollary Forα= ,c=e,we obtain Theorem.of[].
In the following theorem, the recurrence relation, shift operators and differential, integro-differential, partial differential equations are obtained for extended DBernoulli polynomials.
Theorem The extendedD Bernoulli polynomials satisfy the following recurrence rela-tion:
B(α,j)(x,y,c) = , B(–αk,j)(x,y,c) := ,
B(nα+,j)(x,y,c) =
xlnc–α
B(nα,j)(x,y,c) +yj n!
(n–j+ )!(lnc)B
(α,j)
n–j+(x,y,c) ()
– α
n+
n–
k=
n+ k
B(kα,j)(x,y,c)Bn+–k,
Shift operators are given by
L–n:= nlncDx,
L+n:=xlnc–α
+yj(lnc)
(–j)D(j–)
x
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(k–n)Dn–k x ,
L–
n:=
(lnc)j– n D
–j x Dy,
L+
n:=
xlnc–α
+yj(lnc)(j–)(j–)+Dx–(j–)Djy–
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(n–k)(j–)D–(j–)(n–k)
x Dny–k,
where n≥,j≥is an integer and c> .
The differential,integro-differential and partial differential equations for the extended D Bernoulli polynomials are given by
x– α lnc
Dx+yj(lnc)–jDjx
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
k–n–Dn+–k x –n
B(na,j)(x,y,c) = , ()
xlnc–α
Dy+j(lnc)(j–)(j–)+D–(j–)
x Djy–+yj(lnc)(j–)(j–)+D–(j–)
x Djy
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(j–)(n–k)D–(j–)(n–k)
x Dny–k+
– (n+ )(lnc)–jDjx–
B(na,j)(x,y,c) = , ()
xlnc–α
D(xj–)(n–)Dy+ (j– )(n– )D(xj–)(n–)–Dy
+j(lnc)(j–)(j–)+Dx(j–)(n–j)Djy–( +yDy)
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(j–)(n–k)D(j–)(k–)
x Dny–k+– (n+ )(lnc)–jDnx(j–)
×B(na,j)(x,y,c) = , n≥j, ()
respectively.
Proof Taking derivative on both sides of the generating relation
tα (et– )αc
xt+ytj
= ∞
n=
B(α,j)
n (x,y,c)
tn
n!
with respect tot, then using some series manipulations and (), we get the recurrence relation
B(nα+,j)(x,y,c) =
xlnc–α
B(nα,j)(x,y,c) +yj n!
(n–j+ )!(lnc)B
(α,j)
n–j+(x,y,c)
– α
n+
n–
k=
n+ k
B(kα,j)(x,y,c)Bn+–k.
Differentiating generating equation () with respect toxand equating coefficients oftn,
we obtain
DxB(nα,j)(x,y,c) =nlncB
(α,j)
n–(x,y,c).
Hence, the operatorL–n:=nlncDxsatisfies the following relation:
L–nB(nα,j)(x,y,c) =B(nα–,j)(x,y,c).
Since, we have the relations
B(kα,j)(x,y,c) =L–k+L–k+· · ·L–nB(α,j)
n (x,y,c)
=k! n!(lnc)
k–nDn–k x B(
α,j)
n (x,y,c), ()
B(nα–,jj)+(x,y,c) =L–n–j+L–n–j+· · ·L–nB(nα,j)(x,y,c)
=(n–j+ )! n! (lnc)
–jDj–
x B(
α,j)
n (x,y,c), ()
writing () and () in the recurrence relation, we getL+
n
L+n:=xlnc–α
+yj(lnc)
(–j)D(j–)
x –α n–
k=
Bn+–k
(n+ –k)!(lnc)
(k–n)Dn–k x .
By applying the factorization method (see [, ]),
L–n+L+nB(α,j)
n (x,y,c) =B(nα,j)(x,y,c)
we get differential equation
x– α lnc
Dx+yj(lnc)–jDjx
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
k–n–Dn+–k x –n
To obtain the integro-differential equation
xlnc–α
Dy+j(lnc)(j–)(j–)+D–(j–)
x Djy–+yj(lnc)(j–)(j–)+D–(j–)
x Djy
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(j–)(n–k)D–(j–)(n–k)
x Dny–k+– (n+ )(lnc)–jDjx–
×B(na,j)(x,y,c) = ,
we take derivative with respect toyin the generating relation () to obtain
(lnc)B(na–,jj)(x,y,c)n(n– )· · ·(n–j+ ) =∂B
(a,j)
n (x,y,c) ∂y .
From this fact, we writeL–
nas follows:
L–
n:=
(lnc)j–
n D
–j x Dy.
By using this lowering operator in (), we get
L+
n:=
xlnc–α
+yj(lnc)(j–)(j–)+Dx–(j–)Djy–
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(n–k)(j–)D–(j–)(n–k)
x Dny–k.
Using the factorization relation
L–
n+L+nB(nα,j)(x,y,c) =B(nα,j)(x,y,c),
we get the integro-differential equation ().
Differentiating both sides of () with respect tox, (j– )(n– ) times, we obtain the partial differential equation
xlnc–α
D(xj–)(n–)Dy+ (j– )(n– )D(xj–)(n–)–Dy
+j(lnc)(j–)(j–)+Dx(j–)(n–j)Djy–( +yDy)
–α n–
k=
Bn+–k
(n+ –k)!(lnc)
(j–)(n–k)D(j–)(k–)
x Dny–k+– (n+ )(lnc)–jDnx(j–)
×B(na,j)(x,y,c) = .
Corollary For the generalizedD Bernoulli polynomials,the recurrence relation is given
Shift operators are given by
L–n:=
Differential,integro-differential and partial differential equations are
x–α
given in []. For this reason, and for the sake of completeness, we list the recurrence rela-tion, shift operators, differential equations for the two dimensional generalized Bernoulli polynomials in the casec=e,α= ,j= in the following corollary. (Note that the following corollary was recorded in [].)
Corollary Recurrence relation of theD Bernoulli polynomials is written as
B() (x,y) = ,
B()n+(x,y) =
x–
B()n (x,y)
+ nyB()n–(x,y) – n+
n–
k=
n+ k
B()k (x,y)Bn+–k.
Shift operators are given by
L–n:= nDx,
L+n:=x–
+ yDx–
n–
k=
Bn–k+
(n–k+ )!D
n–k x ,
L–
n:=
nD
–
x Dy,
L+
n:=
x–
+ yD–x Dy– n–
k=
Bn–k+
(n–k+ )!D
–(n–k)
x Dny–k.
Differential equation is
x–
Dx+ yDx– n–
k=
Bn+–k
(n+ –k)!D
n+–k x –n
B()n (x,y) = ,
integro-differential equation is given by
x–
Dy+ D–x Dy+ yD–x Dy
–
n–
k=
Bn+–k
(n+ –k)!D
–(n–k)
x Dny–k+– (n+ )Dx
B()n (x,y) = ,
partial differential equation is written as
x–
D(xn–)Dy+ (n– )D(xn–)Dy+ D(xn–)Dy( +yDy)
–
n–
k=
Bn+–k
(n+ –k)!D
(k–)
x Dny–k+– (n+ )Dnx
3 Euler polynomials
In this section, we study the Euler polynomials and the equations satisfied by extended DEuler polynomials. Since the extended DEuler differential equations have not been studied before, the results are new even in the casesc=e,α= . The proof is very similar as in the previous section, therefore, we only exhibit the results.
Theorem The recurrence relation of the extendedD Euler polynomials is given by
E(nα+,j)(x,y,c) =
Shift operators are given by
L–n:=
+α
Since the casec=ereduces to the generalized DEuler polynomials, we thus have the following corollary.
Corollary For the generalizedD Euler polynomials,we have the recurrence:
E(nα+,j)(x,y) =
Differential,integro-differential and partial differential equations:
x–α
4 Concluding remarks
As it is mentioned in the Introduction section, generalized Bernoulli polynomialsB(nα)(x;λ;
a;b;c) and generalized Euler polynomialsEn(α)(x;λ;a;b;c) of orderα∈Cwere constructed
by the following generating relations, respectively []:
tα (λbt–at)αc
xt=
∞
n=
B(α)
n (x;λ;a,b,c)
tn
n!,
α (λbt+at)αc
xt=
∞
n=
En(α)(x;λ;a,b,c)t
n
n!,
wherea,b,c∈R+(a=b)n∈N
. Using factorization method, differential equations can
be obtained for these polynomials.
On the other hand, introducing the two variable polynomial families
tα (λbt–at)αc
xt+ytj=
∞
n=
B(nα,j)(x,y;λ;a,b,c)t
n
n!,
and
α (λbt+at)αc
xt+ytj=
∞
n=
E(α,j)
n (x,j;λ;a,b,c)
tn
n!,
wherea,b,c∈R+(a=b)n∈N
, following the lines of proof given in Section , the
differ-ential, integro-differential and partial differential equations can be given for these families. Future works are left to the interested readers.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
This article is dedicated to Professor Hari M Srivastava.
Received: 5 December 2012 Accepted: 21 March 2013 Published: 17 April 2013 References
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doi:10.1186/1687-1847-2013-107
