Open access peer-reviewed chapter

Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem

By Francesco Aldo Costabile and Elisabetta Longo

Submitted: November 26th 2011Reviewed: April 16th 2012Published: July 11th 2012

DOI: 10.5772/46482

Downloaded: 2037

1. Introduction

In 1880 P. E. Appell ([1]) introduced and widely studied sequences of n-degree polynomials

Anx,n=0,1,...uid1

satisfying the differential relation

DAnx=nAn-1(x),n=1,2,...uid2

Sequences of polynomials, verifying the (), nowadays called Appell polynomials, have been well studied because of their remarkable applications not only in different branches of mathematics ([2], [3]) but also in theoretical physics and chemistry ([4], [5]). In 1936 an initial bibliography was provided by Davis (p. 25[6]). In 1939 Sheffer ([7]) introduced a new class of polynomials which extends the class of Appell polynomials; he called these polynomials of type zero, but nowadays they are called Sheffer polynomials. Sheffer also noticed the similarities between Appell polynomials and the umbral calculus, introduced in the second half of the 19th century with the work of such mathematicians as Sylvester, Cayley and Blissard (for examples, see [8]). The Sheffer theory is mainly based on formal power series. In 1941 Steffensen ([9]) published a theory on Sheffer polynomials based on formal power series too. However, these theories were not suitable as they did not provide sufficient computational tools. Afterwards Mullin, Roman and Rota ([10], [11], [12]), using operators method, gave a beautiful theory of umbral calculus, including Sheffer polynomials. Recently, Di Bucchianico and Loeb ([13]) summarized and documented more than five hundred old and new findings related to Appell polynomial sequences. In last years attention has centered on finding a novel representation of Appell polynomials. For instance, Lehemer ([14]) illustrated six different approaches to representing the sequence of Bernoulli polynomials, which is a special case of Appell polynomial sequences. Costabile ([15], [16]) also gave a new form of Bernoulli polynomials, called determinantal form, and later these ideas have been extended to Appell polynomial sequences. In fact, in 2010, Costabile and Longo ([17]) proposed an algebraic and elementary approach to Appell polynomial sequences. At the same time, Yang and Youn ([18]) also gave an algebraic approach, but with different methods. The approach to Appell polynomial sequences via linear algebra is an easily comprehensible mathematical tool, specially for non-specialists; that is very good because many polynomials arise in physics, chemistry and engineering. The present work concerns with these topics and it is organized as follows: in Section we mention the Appell method ([1]); in Section we provide the determinantal approach ([17]) and prove the equivalence with other definitions; in Section classical and non-classical examples are given; in Section , by using elementary tools of linear algebra, general properties of Appell polynomials are provided; in Section we mention Appell polynomials of second kind ([19], [20]) and, in Section two classical examples are given; in Section we provide an application to general linear interpolation problem([21]), giving, in Section , some examples; in Section the Yang and Youn approach ([18]) is sketched; finally, in Section conclusions close the work.

2. The Appell approach

Let An(x)nbe a sequence of n-degree polynomials satisfying the differential relation (). Then we have

Remark 1 There is a one-to-one correspondence of the set of such sequences An(x)nand the set of numerical sequences αnn,α00given by the explicit representation

Anx=αn+n1αn-1x+n2αn-2x2++α0xn,n=0,1,...uid4

Equation (), in particular, shows explicitly that for each n1the polynomial Anxis completely determined by An-1xand by the choice of the constant of integration αn.

Remark 2 Given the formal power series

ah=α0+h1!α1+h22!α2++hnn!αn+,α00,uid6

with αii=0,1,...real coefficients, the sequence of polynomials, An(x), determined by the power series expansion of the product ahehx, i.e.

ahehx=A0x+h1!A1x+h22!A2x++hnn!Anx+,uid7

satisfies ().

The function ahis said, by Appell, 'generating function' of the sequence An(x)n.

Appell also noticed various examples of sequences of polynomials verifying ().

He also considered ([1]) an application of these polynomial sequences to linear differential equations, which is out of this context.

3. The determinantal approach

Let be βi,i=0,1,...,with β00.

We give the following

Definition 1 The polynomial sequence defined by

A0x=1β0,Anx=-1nβ0n+11xx2xn-1xnβ0β1β2βn-1βn0β021β1n-11βn-2n1βn-100β0n-12βn-3n2βn-200β0nn-1β1,n=1,2,...uid9

is called Appell polynomial sequence for βi.

Then we have

Theorem 1 If Anxis the Appell polynomial sequence for βithe differential relation () holds.

Using the properties of linearity we can differentiate the determinant (), expand the resulting determinant with respect to the first column and recognize the factor An-1xafter multiplication of the i-th row by i-1,i=2,...,nand j-th column by 1j,j=1,...,n.      

Theorem 2 If Anxis the Appell polynomial sequence for βiwe have the equality () with

α0=1β0,αi=-1iβ0i+1β1β2βi-1βiβ021β1i-11βi-2i1βi-10β0i-12βi-3i2βi-200β0ii-1β1==-1β0k=0i-1ikβi-kαk,i=1,2,...,n.uid12

From (), by expanding the determinant Anxwith respect to the first row, we obtain the () with αigiven by () and the determinantal form in (); this is a determinant of an upper Hessenberg matrix of order i([16]), then setting α¯i=(-1)iβ0i+1αifor i=1,2,...,n,we have

α¯i=k=0i-1-1i-k-1hk+1,iqkiα¯k,uid13

where:

hl,m=βmforl=1,ml-1βm-l+1for1<lm+1,0forl>m+1,l,m=1,2,...,i,uid14
qki=j=k+2ihj,j-1=β0i-k-1,k=0,1,...,i-2,qi-1i=1.uid15

By virtue of the previous setting, () implies

α¯i=k=0i-2-1i-k-1ikβi-kβ0i-k-1α¯k+ii-1β1α¯i-1==-1iβ0i+1-1β0k=0i-1ikβi-kαk,

and the proof is concluded.

Remark 3 We note that () and () are equivalent to

k=0iikβi-kαk=1i=00i>0uid17

and that for each sequence of Appell polynomials there exist two sequences of numbers αiand βirelated by ().

Corollary 1 If Anxis the Appell polynomial sequence for βiwe have

Anx=j=0nnjAn-j0xj,n=0,1,...uid19

Follows from Theorem being

Ai0=αi,i=0,1,...,n.uid20

Remark 4 For computation we can observe that αnis a n-order determinant of a particular upper Hessenberg form and it's known that the algorithm of Gaussian elimination without pivoting for computing the determinant of an upper Hessenberg matrix is stable (p. 27[22]).

Theorem 3 If a(h)is the function defined in () and Anxis the polynomial sequence defined by (), setting

β0=1α0,βn=-1α0k=1nnkαkβn-k,n=1,2,...,uid23

we have that An(x)satisfies the (), i.e. An(x)is the Appell polynomial sequence for βi.

Let be

b(h)=β0+h1!β1+h22!β2++hnn!βn+uid24

with βnas in (). Then we have ahbh=1, where the product is intended in the Cauchy sense, i.e.:

ahbh=n=0k=0nnkαkβn-khnn!.

Let us multiply both hand sides of equation

a(h)ehx=n=0Anxhnn!uid25

for 1ahand, in the same equation, replace functions ehxand 1ahby their Taylor series expansion at the origin; then () becomes

n=0xnhnn!=n=0Anxhnn!n=0hnn!βn.uid26

By multiplying the series on the left hand side of () according to the Cauchy-product rules, previous equality leads to the following system of infinite equations in the unknown Anx,n=0,1,...

A0xβ0=1,A0xβ1+A1xβ0=x,A0xβ2+21A1xβ1+A2xβ0=x2,A0xβn+n1A1xβn-1+...+Anxβ0=xn,uid27

From the first one of () we obtain the first one of (). Moreover, the special form of the previous system (lower triangular) allows us to work out the unknown Anxoperating with the first n+1equations, only by applying the Cramer rule:

Anx=1β0n+1β00001β1β000xβ221β1β00x2βn-1n-11βn-2β0xn-1βnn1βn-1nn-1β1xn.

By transposition of the previous, we have

Anx=1β0n+1β0β1β2βn-1βn0β021β1n-11βn-2n1βn-100β0000β0nn-1β11xx2xn-1xn,n=1,2,...,uid28

that is exactly the second one of () after ncircular row exchanges: more precisely, the i-th row moves to the (i+1)-th position for i=1,...,n-1, the n-th row goes to the first position.

Definition 2 The function ahehx, as in () and (), is said 'generating function' of the Appell polynomial sequence Anxfor βi.

Theorems , , concur to assert the validity of following

Theorem 4 (Circular) If Anxis the Appell polynomial sequence for βiwe have

()()()()().
  1. Follows from Theorem .

  2. Follows from Theorem , or more simply by direct integration of the differential equation ().

  3. Follows ordering the Cauchy product of the developments a(h)and ehxwith respect to the powers of hand recognizing polynomials An(x), expressed in form (), as coefficients of hnn!.

  4. Follows from Theorem .

Remark 5 In virtue of the Theorem , any of the relations (), (), (), () can be assumed as definition of Appell polynomial sequences.

4. Examples of Appell polynomial sequences

The following are classical examples of Appell polynomial sequences.

  1. Bernoulli polynomials ([23], [17]):

    βi=1i+1,i=0,1,...,uid38
    a(h)=heh-1;uid39
  2. Euler polynomials ([23], [17]):

    β0=1,βi=12,i=1,2,...,uid41
    a(h)=2eh+1;uid42
  3. Normalized Hermite polynomials ([24], [17]):

    βi=1π-+e-x2xidx=0foriodd(i-1)(i-3)··3·12i2forieven,i=0,1,...,uid44
    a(h)=e-h24;uid45
  4. Laguerre polynomials ([24], [17]):

    βi=0+e-xxidx=Γi+1=i!,i=0,1,...,uid47
    a(h)=1-h;uid48

The following are non-classical examples of Appell polynomial sequences.

  1. Generalized Bernoulli polynomials

    1. with Jacobi weight ([17]):

      βi=01(1-x)αxβxidx=Γα+1Γβ+i+1Γα+β+i+2,α,β>-1,i=0,1,...,uid51
      a(h)=101(1-x)αxβehxdx;uid52
    2. of order k([11]):

      βi=1i+1k,kinteger,i=0,1,...,uid54
      a(h)=heh-1k;uid55
  2. Central Bernoulli polynomials ([25]):

    β2i=1i+1,β2i+1=0,i=0,1,...,a(h)=hsinh(h);uid57
  3. Generalized Euler polynomials ([17]):

    β0=1,βi=w1w1+w2,w1,w2>0,i=1,2,...,a(h)=w1+w2w1eh+w2;uid59
  4. Generalized Hermite polynomials ([17]):

    βi=1π-+e-xαxidx=0foriodd2απΓi+1αforieven,i=0,1,...,α>0,a(h)=π-e-xαehxdx;uid61
  5. Generalized Laguerre polynomials ([17]):

    βi=0+e-αxxidx=Γi+1αi+1=i!αi+1,α>0,i=0,1,...,a(h)=α-h.uid63

5. General properties of Appell polynomials

By elementary tools of linear algebra we can prove the general properties of Appell polynomials.

Let Anx, n=0,1,...,be a polynomial sequence and βi,i=0,1,...,with β00.

Theorem 5 (Recurrence) Anxis the Appell polynomial sequence for βiif and only if

An(x)=1β0xn-k=0n-1nkβn-kAkx,n=1,2,...uid65

Follows observing that the following holds:

Anx=-1nβ0n+11xx2xn-1xnβ0β1β2βn-1βn0β021β1n-11βn-2n1βn-100β0n-12βn-3n2βn-200β0nn-1β1=uid66
=1β0xn-k=0n-1nkβn-kAkx,n=1,2,...uid67

In fact, if Anxis the Appell polynomial sequence for βi, from (), we can observe that An(x)is a determinant of an upper Hessenberg matrix of order n+1([16]) and, proceeding as in Theorem , we can obtain the ().

Corollary 2 If Anxis the Appell polynomial sequence for βithen

xn=k=0nnkβn-kAkx,n=0,1,...uid69

Follows from ().

Corollary 3 Let 𝒫nbe the space of polynomials of degree nand An(x)nbe an Appell polynomial sequence, then An(x)nis a basis for 𝒫n.

If we have

Pn(x)=k=0nan,kxk,an,k,uid71

then, by Corollary , we get

Pn(x)=k=0nan,kj=0kkjβk-jAjx=k=0ncn,kAk(x),

where

cn,k=j=0n-kk+jkak+jβj.uid72

Remark 6 An alternative recurrence relation can be determined from () after differentiation with respect to h([18], [26]).

Let be βi,γi,i=0,1,...,with β0,γ00.

Let us consider the Appell polynomial sequences Anxand Bnx,n=0,1,...,for βiand γi, respectively, and indicate with ABnxthe polynomial that is obtained replacing in Anxthe powers x0,x1,...,xn, respectively, with the polynomials B0x,B1x,...,Bnx.Then we have

Theorem 6 The sequences

  1. λAnx+μBnx,λ,μ,

  2. ABnx

are sequences of Appell polynomials again.

  1. Follows from the property of linearity of determinant.

  2. Expanding the determinant ABnxwith respect to the first row we obtain

    ABnx=-1nβ0n+1j=0n-1jβ0jnjα¯n-jBjx==j=0n-1n-jβ0n-j+1njα¯n-jBjx,uid79

    where

    α¯0=1,α¯i=β1β2βi-1βiβ021β1i-11βi-2i1βi-10β0i-12βi-3i2βi-200β0ii-1β1,i=1,2,...,n.

    We observe that

    Ai0=-1iβ0i+1α¯i,i=1,2,...,n

    and hence () becomes

    ABnx=j=0nnjAn-j0Bjx.uid80

    Differentiating both hand sides of () and since Bjxis a sequence of Appell polynomials, we deduce

    ABnx'=nABn-1x.uid81

Let us, now, introduce the Appell vector.

Definition 3 If Anxis the Appell polynomial sequence for βithe vector of functions A¯nx=[A0(x),...,An(x)]Tis called Appell vector for βi.

Then we have

Theorem 7 (Matrix form) Let A¯nxbe a vector of polynomial functions. Then A¯nxis the Appell vector for βiif and only if, putting

Mi,j=ijβi-jij0otherwise,i,j=0,...,n,uid84

and X(x)=1,x,...,xnTthe following relation holds

X(x)=MA¯nxuid85

or, equivalently,

A¯nx=M-1X(x),uid86

being M-1the inverse matrix of M.

If A¯nxis the Appell vector for βithe result easily follows from Corollary .

Vice versa, observing that the matrix Mdefined by () is invertible, setting

M-1i,j=ijαi-jij0otherwise,i,j=0,...,n,uid87

we have the () and therefore the () and, being the coefficients αkand βkrelated by (), we have that An(x)is the Appell polynomial sequence for βi.

Theorem 8 (Connection constants) Let A¯n(x)and B¯n(x)be the Appell vectors for βiand γi, respectively. Then

A¯n(x)=CB¯n(x),uid89

where

Ci,j=ijci-jij0otherwise,i,j=0,...,n.uid90

with

cn=k=0nnkαn-kγk.uid91

From Theorem we have

X(x)=MA¯nx

with Mas in () or, equivalently,

A¯nx=M-1X(x),

with M-1as in ().

Always from Theorem we get

X(x)=NB¯nx

with

Ni,j=ijγi-jij0otherwise,i,j=0,...,n.uid92

Then

A¯nx=M-1NB¯nx,

from which, setting C=M-1N, we have the thesis.

Theorem 9 (Inverse relations) Let Anxbe the Appell polynomial sequence for βithen the following are inverse relations:

yn=k=0nnkβn-kxkxn=k=0nnkAn-k(0)yk.uid94

Let us remember that

Ak(0)=αk,

where the coefficients αkand βkare related by ().

Moreover, setting y¯n=[y0,...,yn]Tand x¯n=[x0,...,xn]T, from () we have

y¯n=M1x¯nx¯n=M2y¯n

with

M1i,j=ijβi-jij0otherwise,i,j=0,...,n,uid95
M2i,j=ijαi-jij0otherwise,i,j=0,...,n,uid96

and, from () we get

M1M2=In+1,

i.e. () are inverse relations.

Theorem 10 (Inverse relation between two Appell polynomial sequences) Let A¯n(x)and B¯n(x)be the Appell vectors for βiand γi, respectively. Then the following are inverse relations:

A¯n(x)=CB¯n(x)B¯n(x)=C˜A¯n(x)uid98

with

Ci,j=ijci-jij0otherwise,C˜i,j=ijc˜i-jij0otherwise,i,j=0,...,n,uid99
cn=k=0nnkAn-k(0)γk,c˜n=k=0nnkBn-k(0)βk.uid100

Follows from Theorem , after observing that

k=0nnkcn-kc˜k=1n=00n>0uid101

and therefore

CC˜=In+1.

Theorem 11 (Binomial identity) If Anxis the Appell polynomial sequence for βiwe have

Anx+y=i=0nniAixyn-i,n=0,1,...uid103

Starting by the Definition and using the identity

x+yi=k=0iikykxi-k,uid104

we infer

Anx+y=-1nβ0n+11(x+y)1(x+y)n-1(x+y)nβ0β1βn-1βn00β0β1nn-1=
=i=0nyi-1n-iβ0n-i+1iii+1ix1i+2ix2n-1ixn-i-1nixn-iβ0β1i+1iβ2i+2iβn-i-1n-1iβn-ini0β0β1i+2i+1βn-i-2n-1i+1βn-i-1ni+1β000β0β1nn-1.

We divide, now, each j-th column, j=2,...,n-i+1, for i+j-1iand multiply each h-th row, h=3,...,n-i+1, for i+h-2i. Thus we finally obtain

Anx+y==i=0ni+1inii+1in-1iyi-1n-iβ0n-i+11x1x2xn-i-1xn-iβ0β1β2βn-i-1βn-i0β0β121βn-i-2n-i-11βn-i-1n-i1β00......0β0β1n-in-i-1==i=0nniAn-ixyi=i=0nniAixyn-i.

      

Theorem 12 (Generalized Appell identity) Let An(x)and Bn(x)be the Appell polynomial sequences for βiand γi,respectively. Then, if Cn(x)is the Appell polynomial sequence for δiwith

δ0=1C0(0),δi=-1C0(0)k=1iikδi-kCk(0),i=1,...,uid106

and

Ci(0)=j=0iijBi-j(0)Aj(0),uid107

where Ai(0)and Bi(0)are related to βiand γi, respectively, by relations similar to (), we have

Cn(y+z)=k=0nnkAk(y)Bn-k(z).uid108

Starting from () we have

Cn(y+z)=k=0nnkCn-k(0)(y+z)k.uid109

Then, applying () and the well-known classical binomial identity, after some calculation, we obtain the thesis.

Theorem 13 (Combinatorial identities) Let An(x)and Bn(x)be the Appell polynomial sequences for βiand γi,respectively. Then the following relations holds:

k=0nnkAk(x)Bn-k(-x)=k=0nnkAk(0)Bn-k(0),uid111
k=0nnkAk(x)Bn-k(z)=k=0nnkAk(x+z)Bn-k(0).uid112

If Cn(x)is the Appell polynomial sequence for δidefined as in (), from the generalized Appell identity, we have

k=0nnkAk(x)Bn-k(-x)=Cn(0)=k=0nnkAk(0)Bn-k(0)

and

k=0nnkAk(x)Bn-k(z)=Cn(x+z)=k=0nnkAk(x+z)Bn-k(0).

Theorem 14 (Forward difference) If Anxis the Appell polynomial sequence for βiwe have

ΔAnxAnx+1-Anx=i=0n-1niAix,n=0,1,...uid114

The desired result follows from () with y=1.

Theorem 15 (Multiplication Theorem) Let A¯n(x)be the Appell vector for βi.

The following identities hold:

A¯nmx=B(x)A¯nxn=0,1,...,m=1,2,...,uid116
A¯nmx=M-1DX(x)n=0,1,...,m=1,2,...,uid117

where

B(x)i,j=ij(m-1)i-jxi-jij0otherwise,i,j=0,...,n,uid118

D=diag[1,m,...,mn]and M-1defined as in ().

The () follows from () setting y=xm-1. In fact we get

Anmx=i=0nniAixm-1n-ixn-i.uid119

The () follows from Theorem . In fact we get

A¯n(mx)=M-1X(mx)=M-1DX(x),uid120

and

Anmx=i=0nniαn-imixi.uid121

Theorem 16 (Differential equation) If Anxis the Appell polynomial sequence for βithen Anxsatisfies the linear differential equation:

βnn!y(n)(x)+βn-1(n-1)!y(n-1)(x)+...+β22!y(2)(x)+β1y(1)(x)+β0y(x)=xnuid123

From Theorem we have

An+1(x)=1β0xn+1-k=0nn+1k+1βk+1An-k(x).uid124

From Theorem we find that

An+1'(x)=(n+1)An(x),andAn-k(x)=An(k)(x)n(n-1)...(n-k+1),uid125

and replacing An-k(x)in the () we obtain

An+1(x)=1β0xn+1-(n+1)k=0nβk+1An(k)(x)(k+1)!.uid126

Differentiating both hand sides of the last one and replacing An+1'(x)with (n+1)An(x), after some calculation we obtain the thesis.

Remark 7 An alternative differential equation for Appell polynomial sequences can be determined by the recurrence relation referred to in Remark ([18], [26]).

6. Appell polynomial sequences of second kind

Let f:Iand Δbe the finite difference operator ([23]), i.e.:

Δ[f](x)=f(x+1)-f(x),uid128

we define the finite difference operator of order i, with i, as

Δi[f](x)=Δ(Δi-1[f](x))=j=0i(-1)i-jijf(x+j),uid129

meaning Δ0=Iand Δ1=Δ, where Iis the identity operator.

Let the sequence of falling factorial defined by

x0=1,xn=xx-1x-2x-n+1,n=1,2,...,uid130

we give the following

Definition 4 Let bad hboxi, i=0,1,...,with bad hbox00. The polynomial sequence

𝒜0x=1bad hbox0,𝒜nx=-1nbad hbox0n+11x1x2xn-1xnbad hbox0bad hbox1bad hbox2bad hboxn-1bad hboxn0bad hbox021bad hbox1n-11bad hboxn-2n1bad hboxn-100bad hbox0n-12bad hboxn-3n2bad hboxn-200bad hbox0nn-1bad hbox1,n=1,2,...uid132

is called Appell polynomial sequence of second kind.

Then, we have

Theorem 17 For Appell polynomial sequences of second kind we get

Δ𝒜nx=n𝒜n-1xn=1,2,...uid134

By the well-known relation ([23])

Δxn=nxn-1,n=1,2,...,uid135

applying the operator Δto the definition () and using the properties of linearity of Δwe have

Δ𝒜nx=-1n0n+1Δ1Δx1Δx2Δxn-1Δxn012n-1n00211n-11n-2n1n-1000n-12n-3n2n-2000nn-11,n=1,2,...uid136

We can expand the determinant in () with respect to the first column and, after multiplying the i-th row by i-1,i=2,...,nand the j-th column by 1j,j=1,...,n,we can recognize the factor 𝒜n-1x.

We can observe that the structure of the determinant in () is similar to that one of the determinant in (). In virtue of this it is possible to obtain a dual theory of Appell polynomials of first kind, in the sense that similar properties can be proven ([20]).

For example, the generating function is

H(x,h)=a(h)(1+h)x,uid137

where a(h)is an invertible formal series of power.

7. Examples of Appell polynomial sequences of second kind

The following are classical examples of Appell polynomial sequences of second kind.

  1. Bernoulli polynomials of second kind ([23], [20]):

    i=-1ii+1i!,i=0,1,...,uid139
    H(x,h)=h(1+h)xln(1+h);uid140
  2. Boole polynomials ([23], [20]):

    i=1,i=012,i=10,i=2,...uid142
    H(x,h)=2(1+h)x2+h.uid143

8. An application to general linear interpolation problem

Let Xbe the linear space of real functions defined in the interval [0,1]continuous and with continuous derivatives of all necessary orders. Let Lbe a linear functional on Xsuch that L(1)0. If in () and respectively in () we set

βi=L(xi),i=L((x)i),i=0,1,...,uid144

An(x)and 𝒜n(x)will be said Appell polynomial sequences of first or of second kind related to the functional Land denoted by AL,n(x)and 𝒜L,n(x), respectively.

Remark 8 The generating function of the sequence AL,n(x)is

G(x,h)=exhLx(exh),uid146

and for 𝒜L,n(x)is

H(x,h)=(1+h)xLx((1+h)x),uid147

where Lxmeans that the functional Lis applied to the argument as a function of x.

For AL,n(x)if G(x,h)=a(h)exhwith 1a(h)=i=0βihii!we have

G(x,t)=exh1a(h)=exhi=0βihii!=exhi=0L(xi)hii!=exhLi=0xihii!=exhLx(exh).

For 𝒜L,n(x), the proof similarly follows. Then, we have

Theorem 18 Let ωi,i=0,...,n,the polynomials

Pn(x)=i=0nωii!AL,i(x),uid149
Pn*(x)=i=0nωii!𝒜L,i(x)uid150

are the unique polynomials of degree less than or equal to n,such that

L(Pn(i))=i!ωi,i=0,...,n,uid151
L(ΔiPn*)=i!ωi,i=0,...,n.uid152

The proof follows observing that, by the hypothesis on functional Lthere exists a unique polynomial of degree nverifying () and , respectively, (); moreover from the properties of AL,i(x)and 𝒜L,i(x), we have

L(AL,i(j)(x))=i(i-1)...(i-j+1)L(AL,i-j(x))=j!ijδij,uid153
L(Δi𝒜L,i(x))=i(i-1)...(i-j+1)L(𝒜L,i-j(x))=j!ijδij,uid154

where δijis the Kronecker symbol.

From () and () it is easy to prove that the polynomials () and () verify () and (), respectively.

Remark 9 For every linear functional Lon X, {AL,i(x)},{𝒜L,i(x)},i=0,...,n,are basis for 𝒫nand, Pn(x)𝒫n, we have

Pn(x)=i=0nL(Pn(i))i!AL,i(x),uid156
Pn(x)=i=0nL(ΔiPn)i!𝒜L,i(x).uid157

Let us consider a function fX.Then we have the following

Theorem 19 The polynomials

PL,n[f](x)=i=0nL(f(i))i!AL,i(x),uid159
PL,n*[f](x)=i=0nL(Δif)i!𝒜L,i(x)uid160

are the unique polynomial of degree nsuch that

L(PL,n[f](i))=L(f(i)),i=0,...,n,
L(ΔiPL,n*[f])=L(Δif),i=0,...,n.

Setting ωi=L(f(i))i!,andrespectively,ωi=L(Δif)i!,i=0,...,n,the result follows from Theorem .

Definition 5 The polynomials () and () are called Appell interpolation polynomial for fof first and of second kind, respectively.

Now it is interesting to consider the estimation of the remainders

RL,n[f](x)=f(x)-PL,n[f](x),x[0,1],uid162
RL,n*[f](x)=f(x)-PL,n*[f](x),x[0,1].uid163

Remark 10 For any f𝒫n

RL,n[f](x)=0,RL,n[xn+1]0,x[0,1],uid165
RL,n*[f](x)=0,RL,n*[(x)n+1]0,x[0,1],uid166

i. e. the polynomial operators () and () are exact on 𝒫n.

For a fixed xwe may consider the remainder RL,n[f](x)and RL,n*[f](x)as linear functionals which act on fand annihilate all elements of 𝒫n. From Peano's Theorem (p. 69[27]) if a linear functional has this property, then it must also have a simple representation in terms of f(n+1). Therefore we have

Theorem 20 Let fCn+1a,b,the following relations hold

RL,n(f,x)=1n!01Kn(x,t)fn+1tdt,x0,1,uid168
RL,n*(f,x)=1n!01Kn*(x,t)fn+1tdt,x0,1,uid169

where

Kn(x,t)=RL,nx-t+n=x-t+n-i=0nniL(x-t)+n-iAL,i(x),uid170
Kn*(x,t)=RL,n*x-t+n=x-t+n-i=0nLΔi(x-t)+ni!𝒜L,i(x).uid171

After some calculation, the results follow by Remark and Peano's Theorem.

Remark 11 (Bounds) If f(n+1)p[0,1]and Kn(x,t),Kn*(x,t)q[0,1]with 1p+1q=1then we apply the Hölder's inequality so that

RL,n[f](x)1n!01Kn(x,t)qdt1q01fn+1tpdt1p,
RL,n*[f](x)1n!01Kn*(x,t)qdt1q01fn+1tpdt1p.

The two most important cases are p=q=2and q=1,p=:

  1. for p=q=2we have the estimates

    RL,n[f](x)σnf,RL,n*[f](x)σn*f,uid174

    where

    (σn)2=1n!201Kn(x,t)2dt,(σn*)2=1n!201Kn*(x,t)2dt,uid175

    and

    f2=01fn+1t2dt;uid176
  2. for q=1,p=we have that

    RL,n[f](x)1n!Mn+101Kn(x,t)dt,RL,n*[f](x)1n!Mn+101Kn*(x,t)dt,uid178

    where

    Mn+1=supaxbfn+1x.uid179

A further polynomial operator can be determined as follows:

for any fixed z0,1we consider the polynomial

P¯L,n[f](x)f(z)+PL,n[f](x)-PL,n[f](z)=f(z)+i=1nL(f(i))i!AL,i(x)-AL,i(z),uid180

and, respectively,

P¯L,n*[f](x)f(z)+PL,n*[f](x)-PL,n*[f](z)=f(z)+i=1nL(Δif)i!𝒜L,i(x)-𝒜L,i(z).uid181

Then we have the following

Theorem 21 The polynomials P¯L,n[f](x), P¯L,n*[f](x)are approximating polynomials of degree nfor f(x), i.e.:

x0,1,f(x)=P¯L,n[f](x)+R¯L,n[f](x),uid183
f(x)=P¯L,n*[f](x)+R¯L,n*[f](x),uid184

where

R¯L,n[f](x)=RL,n[f](x)-RL,n[f](z),uid185
R¯L,n*[f](x)=RL,n*[f](x)-RL,n*[f](z),uid186

with

R¯L,n[xi]=0,i=0,..,n,R¯L,n[xn+1]0,uid187
R¯L,n*[(x)i]=0,i=0,..,n,R¯L,n*[(x)n+1]0.uid188

x0,1and for any fixed z0,1, from (), we have

f(x)-f(z)=PL,n[f](x)-PL,n[f](z)+RL,n[f](x)-RL,n[f](z),

from which we get () and (). The exactness of the polynomial P¯L,n[f](x)follows from the exactness of the polynomial PL,n[f](x).

Proceeding in the same manner we can prove the result for the polynomial P¯L,n*[f](x).

Remark 12 The polynomials P¯L,n[f](x), P¯L,n*[f](x)satisfy the interpolation conditions

P¯L,n[f](z)=f(z),L(P¯L,n(i)[f])=L(f(i)),i=1,...,n,uid190
P¯L,n*[f](z)=f(z),L(ΔiP¯L,n*[f])=L(Δif),i=1,...,n.uid191

9. Examples of Appell interpolation polynomials

  1. Taylor interpolation and classical interpolation on equidistant points:

    Assuming

    L(f)=f(x0),x0[0,1],uid193

    the polynomials PL,n[f](x)and PL,n*[f](x)are, respectively, the Taylor interpolation polynomial and the classical interpolation polynomial on equidistant points;

  2. Bernoulli interpolation of first and of second kind:

    1. Bernoulli interpolation of first kind ([21], [15]):

      Assuming

      L(f)=01f(x)dx,uid196

      the interpolation polynomials PL,n[f](x)and P¯L,n[f](x)become

      PL,n[f](x)=01f(x)dx+i=1nf(i-1)(1)-f(i-1)(0)i!Bix,uid197
      P¯L,n[f](x)=f(0)+i=1nf(i-1)(1)-f(i-1)(0)i!Bix-Bi0,uid198

      where Bi(x)are the classical Bernoulli polynomials ([17], [23]);

    2. Bernoulli interpolation of second kind ([20]):

      Assuming

      L(f)=DΔ-1fx=0,uid200

      where Δ-1denote the indefinite summation operator and is defined as the linear operator inverse of the finite difference operator Δ, the interpolation polynomials PL,n*[f](x)and P¯L,n*[f](x)become

      PL,n*[f](x)=[Δ-1Df]x=0+i=0n-1f'(i)n,iIIx,uid201
      P¯L,n*[f](x)=f(0)+i=0n-1f'(i)n,iIIx-n,iII0,uid202

      where

      n,iII(x)=j=in-1ji(-1)j-i(j+1)!Bj+1IIx,uid203

      and BjII(x)are the Bernoulli polynomials of second kind ([20]);

  3. Euler and Boole interpolation:

    1. Euler interpolation ([21]):

      Assuming

      L(f)=f(0)+f(1)2,uid206

      the interpolation polynomials PL,n[f](x)and P¯L,n[f](x)become

      PL,n[f](x)=f(0)+f(1)2+i=1nf(i)(0)+f(i)(1)2i!Eix,uid207
      P¯L,n[f](x)=f(0)+i=1nf(i)(0)+f(i)(1)2i!Eix-Ei0;uid208
    2. Boole interpolation ([20]):

      Assuming

      L(f)=Mfx=0,uid210

      where Mfis defined by

      Mf(x)=f(x)+f(x+1)2,uid211

      the interpolation polynomials PL,n*[f](x)and P¯L,n*[f](x)become

      PL,n*[f](x)=f(0)+f(1)2n,0II(x)+i=1nf(i)+f(i+1)2n,iII(x),uid212
      P¯L,n*[f](x)=f(0)+i=1nf(i)+f(i+1)2n,iII(x)-n,iII(0),uid213

      where

      n,iII(x)=j=inji(-1)j-ij!EjII(x),uid214

      and EjII(x)are the Boole polynomials ([20]).

10. The algebraic approach of Yang and Youn

Yang and Youn ([18]) also proposed an algebraic approach to Appell polynomial sequences but with different methods. In fact, they referred the Appell sequence, sn(x), to an invertible analytic function g(t):

sn(x)=dndt1g(t)extt=0,uid215

and called Appell vector for g(t)the vector

S¯n(x)=s0(x),...,sn(x)T.uid216

Then, they proved that

S¯n(x)=Pn1g(t)t=0Wnextt=0=Wn1g(t)extt=0,uid217

being Wnf(t)=f(t),f'(t),...,f(n)(t)Tand Pn[f(t)]the generalized Pascal functional matrix of f(t)([28]) defined by

Pn[f(t)]i,j=ijf(i-j)(t)ij0otherwise,i,j=0,...,n.uid218

Expressing the () in matrix form we have

S¯n(x)=SX(x),uid219

with

S=s00000s10s1100s20s21s220sn0sn1sn2snn,X(x)=1,x,...,xnT,uid220

where

si,j=ij1g(t)(i-j)t=0,i=0,...,n,j=0,...,i.uid221

It is easy to see that the matrix Scoincides with the matrix M-1introduced in Section , Theorem .

11. Conclusions

We have presented an elementary algebraic approach to the theory of Appell polynomials. Given a sequence of real numbers βi,i=0,1,...,β00, a polynomial sequence on determinantal form, called of Appell, has been built. The equivalence of this approach with others existing was proven and, almost always using elementary tools of linear algebra, most important properties od Appell polynomials were proven too. A dual theory referred to the finite difference operator Δhas been proposed. This theory has provided a class of polynomials called Appell polynomials of second kind. Finally, given a linear functional L, with L(1)0, and defined

L(xi)=βi,L((x)i)=i,uid222

the linear interpolation problem

L(Pn(i))=i!ωi,L(ΔiPn)=i!ωi,Pn𝒫n,ωi,uid223

has been considered and its solution has been expressed by the basis of Appell polynomials related to the functional Lby (). This problem can be extended to appropriate real functions, providing a new approximating polynomial, the remainder of which can be estimated too. This theory is susceptible of extension to the more general class of Sheffer polynomials and to the bi-dimensional case.

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Francesco Aldo Costabile and Elisabetta Longo (July 11th 2012). Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem, Linear Algebra - Theorems and Applications, Hassan Abid Yasser, IntechOpen, DOI: 10.5772/46482. Available from:

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