Generalized Exponential Euler Polynomials and
Exponential Splines
Tian-xiao He
Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, USA E-mail: the@iwu.edu
Received April 7, 2011, revised May 4, 2011, accepted May 20, 2011
Abstract
Here presented is constructive generalization of exponential Euler polynomial and exponential splines based on the interrelationship between the set of concepts of Eulerian polynomials, Eulerian numbers, and Eulerian fractions and the set of concepts related to spline functions. The applications of generalized exponential Euler polynomials in series transformations and expansions are also given.
Keywords:Eulerian Polynomial, Eulerian Number, Eulerian Fraction, Exponential Euler Polynomial,
Euler-Frobenius Polynomial, B-Spline, Exponential Spline
1. Introduction
Eulerian polynomial is defined by the following equation (cf. for examples, [1-3]):
10
= , | |
1
n n
n
E u u
u
u < 1. (1.1)
It is well-known that the Eulerian polynomial, En
u , of degree n can be written in the form
0
=1
= n , k, =
n k
E u
E n k u E u 1,
.
(1.2) where are called the Eulerian numbers that can be calculated by using
,
E n k
=0
1
, = k 1 j n 1 ; ,0 = 1
j
n
E n k k j k n E n
j
(1.3) The Eulerian number gives the number of permutations of
with permutation ascents (cf. [4]), or equivalently, the number of per- mutation runs of length (cf. [3]), where a set of ascending sequences in a permutation is called a run and is a permutation ascent if the th term in the permutation less than the
,E n k
,n
1 k
1
j
1, 2, k1
j j
st term. A geometric interpretation of Eulerian number E n k
, were pre-sented in [5] and [6], where which was found equaling to the volume of th slice of a -dimensional unit cube, k n namely,
0,1 :n 1 n=1j j
that the Eulerian fraction is a powerful tool in the study of the Eulerian numbers, Eulerian polynomial, Euler function and its generalization, Jordan function ( cf. [7]). The classical Eulerian fraction, m
x , can be ex- pressed in the form
1:= , = 1.
1
n
n n
E u u
u
u (1.4)
B B
spline functions are probably the most applicable one-dimensional polynomial spline functions. The values of spline functions can be defined as the volumes of slices of a -dimensional cube (cf. [8]). Hence, one can expect a tight connection between the Eulerian numbers and the spline function values. This is is one of the motivations to study the interrelationship between Eule- rian polynomials and B-splines. Another motivation is from the construction of the exponential splines by using the exponential Euler polynomial discussed in [9], which will be described in this paper.
n
B
x k
xPolynomial spline functions can be considered as bro- ken polynomials with certain smoothness, which are used to overcome the stiffness of polynomials, for in- stance the Runge phenomenon. One way to construct the univariate polynomial B-splines is using divided dif- ferences to truncated powers, and the truncated powers can be generated by iterated integrations. An extension called the truncated Tchebycheff functions can be con- structed similarly using the iterated integrations with respect to positive, smooth weight functions, and the Tchebycheffian B-splines are linear combination of these k
truncated Tchebycheff functions. Exponential splines be- long to a class of Tchebycheffian B-splines, which pre- serve many features of the polynomial B-splines such as smoothness, compact support, non-negativity, and local linear independence. One may find more details in [7,10,11] and [12] and some applications in [13] and [14]. In [15], Dyn and Ron considered periodic exponential B-splines defined by weight functions i
with , i , and showed these
B-splines possess a significant property of translation invariant and satisfy a generalized Hermite-Genocchi formula. Motivated by this work, Ron define the higher dimensional -directional exponential box splines in [16].
= ea ui
i
w u r u
1 =
i i
r u r u
n
a
Another approach to construct the exponential splines to the base is using a linear combination of the translations of a polynomial B-spline with combination coefficients
u
j
u . The exponential splines to the base can be presented and evaluated by using exponential Euler polynomials, which connect to some famous generating functions (GF’s) in the combinatorics. The details can be found in Lecture 3 of [9-11], and Chapter 2 of [17], and we shall briefly quote them in next section. This approach can be used to derive a strong inter- relationship between two different fields. For instance, in [18], the author presents the equivalence between exponential Euler polynomials, Euler-Frobenius polyno- mials, etc. and some frequently used concepts in com- binatorics such as Eulerian fractions, Eulerian poly- nomials, etc., respectively, which will be surveyed as follows.
u
Let be an integer, . The symbol n
denotes the class of splines of order , i.e., functions
n n0 S
n
s x
s xsatisfying the following conditions: (1) and (2)
1
n
C s x
πn in each interval ,
1
, 1 ,
j j
= 0
j , 2, , where is the class of polynomials
of degree not exceeding .
πn n
In the central part of Schoenberg’s lectures on “Cardinal Spline Interpolation” (CSI) (cf. [9]), the exponential Euler polynomial of degree to the base
is defined as
n u
; := ! 1
1
n
; , 0 1, = 0,1,n n
A x u n u x u x u (1.5) where
; := 1
jn x u u Qn x
jis the exponential spline of degree to the base , where
n u
:=
;0,1, ,
n
Q x M x n
denote the cardinal forward B-splines. Here, an ex- ponential spline f Sn means an element in
satisfying the functional equation
n
S
1 =
.f x f x (1.6) Obviously, Qn
x Sn1. Therefore, nSn, and itis easy to find n satisfies (1.6).
From [9], n
x u; satisfies n
x1;u
=u
x u;0 < < 1x
and is a polynomial in the interval with the form
; = 1
1 1
lower degree terms!
n n
n x u u x
n
for 0 < < 1x . Hence, An
x u; is a monic polynomial in 0 x 1. From [9], the generating function of
An x u;
is
0
1e = ; .
! e
n xt
n t
n
u t
A x u n
u
(1.7) Using (1.7) we may expand both sides of1e =1 d 1e d
e e
xt x
t t
u u
t x
u u
t ,
and comparing the coefficients of power of in the expansions on two sides, we obtain
t
; = 1
;n n
.A x u nA x u (1.8) If x= 0, then (1.7) reduces to
0
1 =
! e
n n t
n u
a u n
u
,t (1.9)
where an
u = An
0;u . It is easy to see that
=0
; = n n j
n j
j n
A x u a u x
j
.j
(1.10) In particular,
=0
1; = n .
n
j n
A u a
j
u (1.11)Hence, we call an
u the coefficient (polynomials) of the exponential Euler polynomial An
x u;e . In addition, multiplying (1.9) by z
u and com- paring the coefficients of the powers of , we have the
relations u
0
=0
= 1 n = n .
k n
a u ua u a u
k
k (1.12)Jointing (1.11) and (1.12) yields
1; =
0; .n n
A u uA u (1.13) If An
1;u =uAn
0;u
for all integer , then
(1.8) implies
> 0 n
1
1
1 1
1; = 1; = 0; = 0;
n n n n
A u nA u nuA u uA u
.
1; =
0;n n
A u uA u (1.14) for all integer > 0.
In the following, we call
:=
1
n
0; 1
n u an u u An u u
n
(1.15) the Euler-Frobenius polynomial. Since
1
1
1
0; := ! 1 0;
= ! 1 ,
n
n n
n j n
A u n u u
n u u Q j
(1.16)
n u
defined by (1.15) can be written as
1 1
1 =0
1 1 =0
= !
= !
= ! 1 .
n j
n n
n
j n
j n
j n
j
u n Q j u
n Q n j u
n Q j u
(1.17)
From (1.17), we know has zeros, all negative and simple, with n
u
n1
a zero if and only if 1
is. A proof can be found, for example, in [17]. On the other hand, from (1.9) and (1.15) we have
1 0
1
= .
! 1
n n
z n
n
u z
n
e u u
Transferring to , we change the above equa- tion as
z z
10
e = .
!
1 e 1
z n
n
z n
n
u z n
u u
The left-hand side of the above equation can be ex- panded as
1
0 0 0
e = e = 1
!
1 e .
z n
n z
z
n
z
u u
n u
Comparing the right-hand sides of the last two equa- tions yields
1
0
= 1
1
n n
n
u
u u
.
n u
(1.18)
(1.18) implies
1 = 1 1 .
n u nu n u u u
(1.19)
In the introduction, we give the following definition of the Eulerian polynomial of degree , which is similar to (1.18).
n
10
= , | |
1
n n
n
E u u
u
where the Eulerian polynomial, can be written in the form of (1.2) with the Eulerian numbers calculated using (1.3).
n E uIn addition, Eulerian fraction, denoted by n
u , is defined by
1:= , = 1
1
n
n n
E u u
u
u (1.21)
Comparing (1.18) and (1.20), (1.17) and (1.7), and (1.15) and (1.3), [18] gives the following results.
Proposition 1.1 [18] Let n
u and an
u be the polynomials defined by (1.15) and (1.9), respectively, and let Eulerian polynomials En
u
, Eulerian numbers
,E n k , and Eulerian fractions be defined by (1.20), (1.2), and (1.21), respectively. Denote the car- dinal forward B-spline of order by . Then we can set the relationship between the concepts as
u n an Q un
= 1 if = 0 =
if > 0,
n n
n
u n
E u
u u n
(1.22)
, = ! n 1
> 0E n k n Q k n
, (1.23)
1 = 1 if
= 1 .
1 if >
n
n n
n
u u n
a u u
u n
u
= 0
0 (1.24)
As examples of the interrelationships shown in (1.22) - (1.24), we see
1)
n0
0 u = 1, E u0
=1;2)
n 1
1 u = 1, E u1
=u;3)
22 2
2 = 1 , =
n u u E u uu ; etc. and
1)
n1
E 1,k
= 0,1,0 ,
Q k2
= 0,1,0 ;
2)
3
1 1
2 2, = 0,1,1, 0 , = 0, , , 0 ;
2 2
n E k Q k
3)
4
1 2 1
3 3, = 0,1, 4,1, 0 , = 0, , , ,0
6 3 6
n E k Q k
;
etc.
Using Proposition 1.1, [18] derives many properties of B-spline and Euler-Frobenius polynomials readily from the corresponding properties of Eulerian polynomials and Eulerian numbers, and vice versa. Applications in B-spline interpolation and evaluation of Riemann Zeta function values at odd integers are also given in [18].
Let . It is well-known that the Eulerian fraction defined by (1.21) can be written as
= 0,1
u
1=0
= ! ,
1
j n
n j
j
u u j S n j
u
< 1,
,
where S n j
, are the Stirling numbers of the second j Skind, i.e.,
or equivalently, the=0
! , = j n
t n j t
number of partitions of distinct elements in blocks.
n j
The connection shown in Proposition 1.1 also provides some important applications, shown in Section 2, of B-splines and exponential B-splines to the enumerative combinatorics. In the section, a generalized Eulerian fraction and a generalized exponential spline to the base
are derived. Section 3 discusses the applications of the generalized exponential Euler polynomials in the series transformations and expansions. Furthermore, using the combinatorial technique, we may generate exponential splines in higher dimensional setting, which will be briefly described at the end of the section and will be discussed in a later paper on the multivariate spline functions.
u
2. Generalized Exponential Splines
In this section, we will introduce a generalized Eulerian fraction, from which a generalized exponential spline to the base will be derived. We shall also discuss the corresponding generalized Euler-Frobenius polynomials and their applications.
u
It is obvious that (1.25) is equivalently
1
=0
= n , 1 j j,
n j
u S n j u u
(2.1)where . Hence, we may extend the Eulerian fraction to a more general case, which will be denoted by
= 0,1 u
u
n and called generalized Eulerian fractions
associated with an infinitely differentiable function u g
g u :
=0:= n , j j
n
j
u g u S n j g u u
(2.2)Therefore, n
u n
u
1 u
1
. An analogue of the coefficients of the exponential Euler polynomialsdefined by (1.10) is extended to
n a u
1 = 1 if = 0
:= ,
1
1 if
1
n
n
n n
u g u n
g u a u g u
u g u n g u
> 0
(2.3) which is associated with an analytic function g u
.We now find the generating function of
n u g u
and an
u g u
.Theorem 2.1 Let g u
be an infinitely differentiablefunction, and let n
u g u
and an
u g u
be defined by (2.2) and (2.3), respectively. Then their exponential generating functions are
0 e =
!
n t
n n
t
g u u g u
n
(2.4) and
0
e 1
= ,
1 !
t n
n n
g u t
a u g u
g u n
(2.5)where u
u g u:
=1 .
Proof. Substituting (2.2) into the right-hand side of (2.4) and using the generalized Euler series transform formula (cf. [19] and (3.1) in [20]), we can evaluate the exponential generating function of n
u g u
for- mally as
0 0 =0
=0
0 =0 0 0
0 0 0
= ,
! !
= =
! !
= = 0
! ! !
n n n
j j
n
n n j
j n n j
n
j j n j
u
n j n j
n j j
j j
n jt
j n j
t t
u g u S n j g u u
n n
u t u
g u u j g u
j n
t u u
j g u e g g ue
n j j
! !
= .
n
t
t j n
Similarly, by substituting (2.3) into the right-hand side of (2.5) and applying formula (2.4) to the resulting expression, we have
0
1 =0
0 =0 ! 1
= 1 1
1 !
1
= 1 1
( ) 1 ! 1
e 1
= ,
1 1
n n
n
n n
n n n j
n n
n n n j
t
t a u g u
n
t u g u
g u n
g u t
u g u
g u n g u
g u g u g u
which yields (2.5) and completes the proof of the theorem.
From an
u g u
we now construct polynomials
;
A x u g u for 0 x 1 using the following expan- sion.
0
e 1
e = ; .
1 !
t n
xt n n
g u t
A x u g u
g u n
(2.6)Thus, A x u g un
;
can be evaluated from
n
a u g u as follows.
=0
; = n n j
n j
j n
A x u g u a u g u x
j
1
for 0 x . In particular, An(0;u g u
=an
u g u
and
=0
1; = n .
n j
j
n
A u g u a u g u
j
We call A x u g un
;
the generalized exponential Euler polynomial of degree n associated with g to the base u, and aj
u g u
the coefficient of
;
n
A x u g u . In addition, we call An
x; 1 g
1
the generalized Euler polynomials, and An
x; 1 , theregular Euler polynomials, are the special case of
; 1 1
nA x g for g u
= 1 1
u
.A generalized exponential spline function to the base , denoted by
u n
x u g u;
, is therefore defined using
;
n
A x u g u (0 x 1) as follows.
1
; = 1 ;
!
n
n x u g u g u An x u g u
n
(2.8)
in 0 x 1. However, if one wants to extend
;
n x u g u
to all real x associated with g u
such that n 1,C n
An
x u g u;
needs to satisfysome periodic conditions. Let us consider the class of
1= 1
g u au with real constants and a= 0 . Then we have the following result.
Theorem 2.2 Let function n
x u g u;
be defined by (2.8) on 0 x 1 associated with g u
= 1au
1, where a and are real constants and a= 0 , n can be extended to a continuous function for all real
1
n
C
x if and only if = 0. Furthermore, the extension is fulfilled by the functional equation
1
1
1 1; 1 = ; 1 .n x u au au n x u au
(2.9)
Proof. From [9], for (2.9) combined with the condition that
;
1
1
n1
n x u au C
require
1; 1
n
A x u au
to satisfy in
0,1 the periodic conditions
1
1
1; 1 = 0; 1
n n
A u au auA u au (2.10) for = 0,1, , n1. From (2.7), we have
=0 ! ; = ! ! = ; . ! n n j n j j n n nA x u g u a u g u x
j n
n
A x u g u n
Hence, condition (2.10) is equivalent to
1 1 11; 1 = 0; 1
= 1
n n
n
A u au auA u au
aua u au
(2.11)
for all n1. From (2.7) (2.3), and (2.2), we have
1 1 1 =1 =0 1 1 =1 = 1 1 =1 =1 = 1; 1 ! 1= 1 1 ,
1 1 1
! 1
= 1 1 ,
1 1 1
! 1
= 1 1 ,
1 1 1
1 1 ,
n k j n j k j k k n n j k k j k
k n n k k n n j
k j k
A u au
k au n
S j k j
qu au
k au n
S j k j
qu au
k au S n k
qu au
n
k S j k j
1! 1 , 1 k k k au
au
(2.12) where the last step is due to the identities
1, 1 =
, 1
, 1
S j k S j k k S j k and
=
1 1, 1 = 1 ,
n
j n
j k
n
S j k S n k
j
.The double sum in the rightmost expression of (2.12) can be further changed to
1 =1 = 1=2 = 1
1 =1 = 2 =1 =1 = !
1 1 , 1
1 !
= 1 ,
1
! 1
= 1 ,
1 1 ,1 1 1 = k n n j k k j k
k n n
j
k k j k
k n n
j
k k j k
n
j
j
n n
k j k
k au n
k S j k
j au
k au n
S j k
j au
k au n
au
S j k j au au n au S j j au n au j au
1 1 ! 1 , 1 1 , 1 k j k k au S j kau au
where the last step comes from S j
,1 ll j1. tituting the last expression of the double sum= 1 for a Subs
1=1 =
!
1 1 , 1
1
k j
n n
k k j k
k au n
k S j k
j au
equation
1 =1 = 1 =1 1 =1 = ! 1 , 1 != 1 ,
1
!
1 1 , 1 ,
1
k n n
j
k k j k
k n n k k k n n j k k j k
k au n
S j k
j au
k au S n k
au
k au n
k S j k
j au
which comes from (2.12), and solving the resulting equa-
tion for
1=1 = ! 1 , 1 k j n n k k j k
k au n
S j k
j au
, weobtain
1 =1 = 1 1 =1 ! 1 , 1 != 1 ,
1 1
k n n
j
k k j k
k n n k k k au n
S j k
j au
k au au
au S n k
au au
. (2.13)Substituting the last equation to the first equation of (2.12) yields
1 1 1 =1 1 1 1 1; 1 != 1 1 ,
1 1 1
1 1
= 1 0; 1 ,
1 1 n k n n k k n
A u au
k au au
S n k
au au
au au au
auA u au au
where the last step is implied by
1 1 1 =10; 1 = 1
1 !
= ,
1 1 1
n n
n k
n
k k
A u au a u au
k au S n k
au au
1 (see (2.2) and (2.3). Therefore, if
1
1
1; 1 = 0; 1
n n
A u au auA u au , then
= 0
. The sufficiency is obvious. This completes the proof of the theorem.
Remark 3.1 Other generalized exponential splines are planed to be constructed. And the properties of the exponential splines need to be investigated.
3. Applications in Series Transformations
and Expansions
All formulas presented in this section are formal iden-
tities in which we always assume that x= 1 or x= 1 according as
1x
1 or
1x
1 appears in the formulas.Theorem 3.1 Let g u
and be infinitely diffe- rentiable on
h u
0,
, and let Ak
x u g u;
(0 x 1) be the generalized exponential Euler polynomial to the base u defined by (2.6) or (2.7). Then we have formally
=1 =1 0 0 ! 1= 1 0 0;
! k k k k k k k u h k h g
k
,
g u h A u g
k
u is (3.1)where u not any poles of g u
and Ak
0;u g
for any
uk1.
Proof. First, we apply the operator g uE
= 0
t
to the infinitely differentiable function at , where
is the shift operator.
h t E
=0 =0 =0 =0 1 = 0 != 0 .
! k k t t k k k k
g uE h t g uE h t
k
u h k g
k
On the other hand, we can present
=0 =0 =0 =0 =0 =0 =0 =0 = 1 = 0 != 0 0
! !
1
= 0 0 .
! ! D t t j j D t j j k j k j k j
j k k
k j g uE h t g ue h t
g ue h t
j
u j
g h
j k
x
g j h
j k
By applying (3.1) in [19] to the inner sum of the rightmost side of the above equation for
= kf t t and noting
=0
, = j k
t
S k j t j!, we obtain
=0 =0 =0 =0 =0 =0 =0 =1 1 = 0 ! ! 1= ( , ) 0
! 1
= 0
!
1
= 0 1 0
! t j k j k j k t k j k
j j k
k j k k k k k k k
g uE h t
u
t g u h
j k
S k j g u u h k u g u h
k
h g u g u h a u g u
k
,which implies (3.1) by writing
0
0 = 0 k 0 k
k
h g u
h g u k! formally, where forThis completes the proof of the theorem.
Remark 4.1 The series transformation formulas (3.1) have numerous applications by setting different infinitely differentiable functions for g x
. For instance, If
1 = 1g u u , then (3.1) becomes to
1
=1 =1
0 = 0 0;
1
k k
k
k
k k
u
h k h u h u
u
, (3.2)where u= 1 and the relation (1.5) has been used. Evidently, when is a polynomial, formulas (3.1) and (3.2) become closed form of summation formulas with a finite number of terms. Moreover, the Right-hand side of each formula may also be viewed as a for the sequence of coefficeients contained in the power series on the left-hand side. For example, if
h uGF
=h u u, noting
1
=1
0; = k j ,
k k
j
u u Q j
we have
2 =1= .
1
k
k
u ku
u
At the end of this section, we will show a way to extend our construction of exponential splines to the higher dimensional setting. First, we shall adopt the multi-index notational system. Denote
1
1ˆ , , r , , ,ˆ r ,
t t t x x x
1
1 1
ˆ = r ˆ ˆ , r
t t t t x t x ,t xr ,
1
ˆ
ˆ , , r , 0 0, , 0 ,
u u u
1 ,
r i i i g u
g u
ˆ
1
e 1
1
,
1 1
t t
i i r i
i i
g u g ue
g u g u
=1
ˆ
ˆ r i i,
i x t
x tMoreover, we write 1 1 rr
t t t with
being non-negative integers. Also, means
1, , r ,r ˆ0
i0
i= 1, , r
, and meansi i
for all i= 1,,r.
We now define a generalized higher dimensional ex- ponential polynomials A
x u g u;
as follows.Definition 3.2 Let g ui
i be any given formal power series over the complex number field with gi
0 = 1.Then A
x u g u;
defined by
ˆ
ˆ ˆ
ˆ, ˆ,
1
0
e 1
e 1
e e
1 1
ˆ ˆ , ,
!
t t
i i r
x t x
i
i i
g u g u
g u g u
t A x u g u
t
(3.3)
are called the generalized higher dimensional expon- ential Euler polynomials of degree associated with g to the base , where uˆ uˆ
0,1
r. In particular, for
= 1 1
g u u , (3.3) defines higher dimensional expon- ential Euler polynomials A
x uˆ ˆ; of degree to thebase uˆ by
ˆ ˆ =1 | |ˆ
0
1 ˆ ˆ
e = ; .
! e
r i x t
i t
i
u t
A x u u
(3.4)Let uˆ =
1, 1, , 1
. Then the polynomial
ˆ; 1, , 1
A x defined by (3.4) is called the higher dimensional (regular) Euler polynomial of degree . The more details will be given in a later paper.
4. Acknowledgments
The author greatly appreciate the discussion with Carl deBoor since 2005 and the valuable suggestions and the lecture [17] he offered fervently.
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