Integral Representations and New Generating Functions of Chebyshev Polynomials

Integral representations and new generating functions of Chebyshev polynomials

Clemente Cesarano

Faculty of Engineering, International Telematic University UNINETTUNO Corso Vittorio Emanuele II, 39 – 00186 – Roma, Italy email: [email protected]

ABSTRACT – In this paper we use the two-variable Hermite polynomials and their operational rules to derive integral representations of Chebyshev polynomials. The concepts and the formalism of the Hermite polynomials Hn(x, y) are a powerful tool to obtain most of the properties of the Chebyshev polynomials. By using these results, we also show how it is possible to introduce relevant generalizations of these classes of polynomials and we derive for them new identities and integral representations. In particular we state new generating functions for the ﬁrst and second kind Chebyshev polynomials.

Keywords: Chebyshev polynomials, Integral representations, Generating functions.

2000 Mathematics Subject Classiﬁcation: 33C45, 33D45.

1 Introduction

The Hermite polynomials [1] can be introduced by using the concept and the formalism of the generating function and related operational rules. In the following we recall the main deﬁnitions and properties.

(2) Definition – The two-variable Hermite Polynomials Hm (x, y) of Kampéde Férietform [2], [3] are defined by the following formula:

[ m ] X2 m! H(2)(x, y) = ynxm−2n (1.1) m n!(m − 2n)! n=0

1 In the following we will indicate the two-variable Hermite polynomials of (2) Kamp´ede F´erietform by using the symbol Hm(x, y) instead than Hm (x, y).

The two-variable Hermite polynomials Hm(x, y) are linked to the ordinary Hermite polynomials by the following relations: µ ¶ 1 H x, − = He (x) m 2 m where:

[ m ] X2 (−1)rxn−2r He (x) = m! m r!(n − 2r)!2r r=0 and:

Hm (2x, −1) = Hm(x) where:

[ m ] X2 (−1)r(2x)n−2r H (x) = m! m r!(n − 2r)! r=0 and it is also important to note that the Hermite polynomials Hm(x, y) satisfy the relation: m Hm(x, 0) = x . (1.2)

Proposition 1 – The polynomials Hm(x, y) solve the following partial diﬀer- ential equation: ∂2 ∂ H (x, y) = H (x, y) (1.3) ∂x2 m ∂y m Proof – By deriving, separately with respect to x and to y, in the (1.1), we obtain: ∂ H (x, y) = mH (x, y) ∂x m m−1 ∂ H (x, y) = H (x, y). ∂y m m−2 From the ﬁrst of the above relation, by deriving again with respect to x and by noting the second relation, we end up with the (1.3). The Proposition 1 help us to derive an important operational rule for the

Hermite polynomials Hm(x, y). In fact, by considering the diﬀerential equation

2 (1.3) as linear ordinary in the variable y and by remanding the (1.2) we can immediately state the following relation:

2 y ∂ m Hm(x, y) = e ∂x2 x (1.4)

The generating function of the above Hermite polynomials can be state in many ways, we have in fact

Proposition 2 – The polynomials Hm(x, y) satisfy the following diﬀerential diﬀerence equation: d Y (z) = a nY (z) + b n(n − 1)Y (z) (1.5) dz n n−1 n−2 Yn(0) = δn,0 where a and b are real numbers. Proof – By using the generating function method, by putting:

X+∞ tn G (z; t) = Y (z) n! n n=0 with t continuous variable,we can rewrite the (1.5) in the form: d ¡ ¢ G (z; t) = at + bt2 G (z; t) dz G (0; t) = 1 that is a linear ordinary diﬀerential equation and then its solution reads: ¡ ¢ G (z; t) = exp xt + yt2 where we have putted az = x and bz = y. Finally, by exploiting the r.h.s of the previous relation we ﬁnd the thesis and also the relation linking the Hermite polynomials and their generating function:

¡ ¢ X+∞ tn exp xt + yt2 = H (x, y). (1.6) n! n n=0 The use of operational identities, may signiﬁcantly simplify the study of Her- mite generating functions and the discovery of new relations, hardly achievable by conventional means. By remanding that the following identity µ ¶ 2 n − 1 d n d e 4 dx2 (2x) = 2x − (1) (1.7) dx

3 is linked to the standard Burchnall identity, we can immediately state the following relation.

Proposition 3 – The operational deﬁnition of the polynomials Hn(x) reads

2 − 1 d n e 4 dx2 (2x) = Hn(x) (1.8)

Proof – By exploiting the r.h.s of the (1.7), we immediately obtain the Burchnall identity µ ¶ d n Xn 1 ds 2x − = n! (−1)s H (x) (1.9) dx (n − s)!s! n−s dxs s=0 after using the decoupling Weyl identity, since the commutator of the operators of l.h.s. is not zero. The derivative operator of the (1.9) gives a not trivial contribution only in the case s = 0 and then we can conclude with: µ ¶ d n 2x − (1) = H (x) , dx n which prove the statement. The Burchnall identity can be also inverted to give another important relation for the Hermite polynomials Hn(x). We ﬁnd in fact

Proposition 4 – The polynomials Hn(x) satisfy the following operational identity: µ ¶ µ ¶ 1 d Xn n ds H x + = (2x)n−s (1.10) n 2 dx s dxs s=0 tn Proof – By multiplying the l.h.s. of the above relation by n! and then summing up, we obtain µ ¶ X+∞ n t 1 d 2 x+ 1 d t−t2 H x + = e ( 2 )( dx ) . n! n 2 dx n=0 By using the Weyl identity, the r.h.s. of the above equation reads

2 x+ 1 d t−t2 2xt t d e ( 2 )( dx ) = e e dx and from which the (1.10) immediately follows, after expanding the r.h.s and by equating the like t-powers. The previous results can be used to derive some addition and multiplication relations for the Hermite polynomials.

4 Proposition 5 – The polynomials Hn(x) satisfy the following identity ∀ n, m ∈ N: min(n,m) µ ¶µ ¶ X n m H (x) = (−2)s s!H (x)H (x). (1.11) n+m s s n−s m−s s=0 Proof – By using the Proposition 3, we can write µ ¶ µ ¶ µ ¶ d n d m d n H (x) = 2x − 2x − = 2x − H (x) n+m dx dx dx m and by exploiting the r.h.s. of the above relation, we ﬁnd µ ¶ Xn n ds H (x) = (−1)s H (x) H (x). n+m s n−s dxs m s=0 After noting that the following operational identity holds ds 2sm! H (x) = H (x) dxs m (m − s)! m−s we obtain immediately our statement. From the above proposition we can immediately derive as a particular case, the following identity

Xn (−1)s [H (x)]2 H (x) = (−1)n2n(n!)2 s . (1.12) 2n 2s(s!)2(n − s)! s=0 The use of the identity (1.10), stated in Proposition 4, can be exploited to obtain the inverse of relation contained in the (1.12). We have indeed

2 Proposition 6 – Given the Hermite polynomial Hn(x), the square [Hn(x)] can be written as Xn H (x) H (x)H (x) = [H (x)]2 = 2n(n!)2 2n . (1.13) n n n 2s(s!)2(n − s)! s=0 Proof – We can write · µ ¶ µ ¶¸ 2 2 − 1 d 1 d 1 d [H (x)] = e 4 dx2 H x + H x + . n n 2 dx n 2 dx

By using the relation (1.10), we ﬁnd, after manipulating the r.h.s. " # n 2n 2 X 2 − 1 d n 2 (2x) [H (x)] = e 4 dx2 2 (n!) n 2s(s!)2(n − s)! s=0 and then, from the Burchnall identity (1.7), the thesis.

5 A generalization of the identities stated for the one variable Hermite polynomials can be easily done for the polynomials Hn(x, y). We have in fact

Proposition 7 – The following identity holds µ ¶ µ ¶ ∂ n Xn n ∂s x + 2y (1) = (2y)s H (x, y) (1). (1.14) ∂x s n ∂xs s=0

tn Proof – By multiplying the l.h.s. of the above equation by n! and then summing up, we ﬁnd µ ¶ X+∞ n n t ∂ t x+2y ∂ x + 2y = e ( ∂x )(1). n! ∂x n=0 By noting that the commutator of the two operators of the r.h.s. is · ¸ ∂ tx, t2y = −2t2y ∂x we obtain µ ¶ X+∞ n n t ∂ xt+yt2 2ty ∂ x + 2y = e e ∂x (1). (1.15) n! ∂x n=0 After expanding and manipulating the r.h.s. of the previous relation and by equating the like t powers we ﬁnd immediately the (1.14).

By using the Proposition 6 and the deﬁnition of polynomials Hn(x, y), we can derive a generalization of the Burchnall-type identity: µ ¶ 2 n y ∂ n ∂ e ∂x2 x = x + 2y , (1.16) ∂x and the related inverse: µ ¶ µ ¶ ∂ Xn n ∂s H x − 2y , y = (−2y)s xn−s . (1.17) n ∂x s ∂xs s=0 We can also generalize the multiplication rules obtained for the Hermite polynomials Hn(x), stated in Proposition 5.

Proposition 8 – Given the Kamp´ede F´erietHermite polynomials Hn(x, y). We have

min(n,m) X H (x, y)H (x, y) H (x, y) = m!n! (2y)s n−s m−s . (1.18) n+m (n − s)!(m − s)!s! s=0

6 Proof – By using the relations stated in the (1.14) and (1.16), we can write: µ ¶ ∂ n H (x, y) = x + 2y H (x, y) , n+m ∂x m and then µ ¶ Xn n ∂s H (x, y) = (2y)s H (x, y) H (x, y). (1.19) n+m s n ∂xs m s=0 By noting that ∂s m! xm = xm−2s ∂xs (m − 2s)! we obtain ∂s m! H (x, y) = H (x, y) ,. ∂xs m (m − s)! m−s After substituting the above relation in the (1.19) and rearranging the terms we immediately obtain the thesis. From the previous results, it also immediately follows

min(n,m) X H (x, y) H (x, y)H (x, y) = n!m! (−2y)s n+m−2s . (1.20) n m (n − s)!(m − s)!s! s=0 The previous identity and the equation (1.18) can be easily used to derive the particular case for n = m. We have in fact

Xn [H (x, y)]2 H (x, y) = 2n(n!)2 s , (1.21) 2n (s)!2(n − s)!2s s=0

Xn (−1)sH (x, y) [H (x, y)]2 = (−2y)n(n!)2 2s . (1.22) n (n − s)!(s!)22s s=0 Before concluding this section we want prove two other important relations satisﬁed by the Hermite polynomials Hn(x, y).

Proposition 9 – The Hermite polynomials Hn(x, y) solve the following diﬀer- ential equation ∂2 ∂ 2y H (x, y) + x H (x, y) = nH (x, y) (1.23) ∂x2 n ∂x n n Proof – By using the results derived from the Proposition 7, we can easily write that: µ ¶ ∂ x + 2y H (x, y) = H (x, y) ∂x n n+1

7 and from the recurrence relations ∂ H (x, y) = nH (x, y) ∂x n n−1 we have µ ¶ µ ¶ ∂ ∂ x + 2y H (x, y) = nH (x, y) , ∂x ∂x n n which is the thesis. From this statement can be also derived an important recurrence relation. In fact, by noting that ∂ H (x, y) = xH (x, y) + 2y H (x, y) (1.24) n+1 n ∂x n and then we can conclude with

Hn+1(x, y) = xHn(x, y) + 2nyHn−1(x, y). (1.25)

2 Integral representations of Chebyshev polynomials

In this section we will introduce new representations of Chebyshev polynomials [4], [5], by using the Hermite polynomials and the method of the generating function. Since the second kind Chebyshev polynomials Un(x) reads sin[(n + 1) arccos(x)] Un(x) = √ , (2.1) 1 − x2 by exploiting the right hand side of the above relation, we can immediately get the following explicit form

[ n ] X2 (−1)k(n − k)!(2x)n−2k U (x) = . (2.2) n k!(n − 2k)! k=0

Proposition 10 – The second kind Chebyshev polynomials satisfy the following integral representation [6]:

Z +∞ µ ¶ 1 −t n 1 Un(x) = e t Hn 2x, − dt. (2.3) n! 0 t

8 Proof – By noting that: Z +∞ n! = e−ttndt 0 we can write: Z +∞ (n − k)! = e−ttn−kdt. (2.4) 0

From the explicit form of the Chebyshev polynomials Un(x), given in the (2.1), and by recalling the standard form of the two-variable Hermite polynomials: [ n ] X2 ykxn−2k H (x, y) = n! n k!(n − 2k)! k=0 we can immediately write:

Z [ n ] +∞ X2 (−1)kt−k(2x)n−2k U (x) = e−ttn dt n k!(n − 2k)! 0 k=0 and then the thesis. By following the same procedure we can also obtain an analogous integral representation for the Chebyshev polynomials of ﬁrst kind Tn(x), since their explicit form is given by

[ n ] n X2 (−1)k(n − k − 1)!(2x)n−2k T (x) = . (2.5) n 2 k!(n − 2k)! k=0 By using the same relations written in the previous proposition, we have:

Z +∞ µ ¶ 1 −t n−1 1 Tn(x) = e t Hn 2x, − dt. (2.6) 2(n − 1)! 0 t In the previous Section we have stated some useful operational results re- garding the two-variable Hermite polynomials; in particular we have derived their fundamental recurrence relations. These relations can be used to state important results linking the Chebyshev polynomials of the ﬁrst and second kind.

Theorem 1 – The Chebyshev polynomials Tn(x) and Un(x) satisfy the following recurrence relations d U (x) = nW (x) (2.7) dx n n−1 n U (x) = xW (x) − W (x) n+1 n n + 1 n−1

9 and

Tn+1(x) = xUn(x) − Un−1(x) (2.8) where Z +∞ µ ¶ 2 −t n+1 1 Wn(x) = e t Hn 2x, − dt. (n + 1)! 0 t

Proof – The recurrence relations for the standard Hermite polynomials Hn(x, y) stated in the ﬁrst Section, can be costumed in the form · µ ¶ ¸ µ ¶ µ ¶ 1 ∂ 1 1 (2x) + − H 2x, − = H 2x, − (2.9) t ∂x n t n+1 t µ ¶ µ ¶ 1 ∂ 1 1 H 2x, − = nH 2x, − . 2 ∂x n t n−1 t

From the integral representations stated in the relations (2.3) and (2.6), relevant to the Chebyshev polynomials of the ﬁrst and second kind, and by using the second of the identities written above, we obtain

Z +∞ µ ¶ d 2n −t n 1 Un(x) = e t Hn−1 2x, − dt (2.10) dx n! 0 t and Z +∞ µ ¶ d n −t n−1 1 Tn(x) = e t Hn−1 2x, − dt. (2.11) dx (n − 1)! 0 t It is easy to note that the above relation gives a link between the polynomials

Tn(x) and Un(x); in fact, since

Z +∞ µ ¶ 1 −t n−1 1 Un−1(x) = e t Hn−1 2x, − dt (n − 1)! 0 t we immediately obtain d T (x) = nU (x). (2.12) dx n n−1 By applying the multiplication operator to the second kind Chebyshev polynomials, stated in the ﬁrst of the identities (2.9), we can write

Z +∞ · µ ¶ ¸ µ ¶ 1 −t n+1 1 ∂ 1 Un+1(x) = e t (2x) + − Hn 2x, − dt (n + 1)! 0 t ∂x t that is:

Un+1(x) = (2.13) Z +∞ µ ¶ Z +∞ µ ¶ 2 −t n+1 1 n 2 −t n 1 = x e t Hn 2x, − dt− e t Hn−1 2x, − dt. (n + 1)! 0 t n + 1 n! 0 t

10 The second member of the r.h.s. of the above relation suggest us to introduce the following polynomials Z +∞ µ ¶ 2 −t n+1 1 Wn(x) = e t Hn 2x, − dt (2.14) (n + 1)! 0 t recognized as belonging to the families of the Chebyshev polynomials. Thus, from the relation (2.10), we have:

d U (x) = nW (x) (2.15) dx n n−1 and, from the identity (2.13), we get: n U (x) = xW (x) − W (x). (2.16) n+1 n n + 1 n−1 Finally, by using the multiplication operator for the ﬁrst kind Chebyshev polynomials, we can write Z +∞ · µ ¶ ¸ µ ¶ 1 −t n 1 ∂ 1 Tn+1(x) = e t (2x) + − Hn 2x, − dt (2.17) 2n! 0 t ∂x t and then, after exploiting the r.h.s. of the above relation, we can ﬁnd

Tn+1(x) = xUn(x) − Un−1(x) (2.18) which completely prove the theorem.

3 Generating functions

By using the integral representations and the related recurrence relations, stated in the previous Section, for the Chebyshev polynomials of the ﬁrst and second kind, it is possible to derive a slight diﬀerent relations linking these polynomials and their generating functions [4]–[6].

We note indeed, for the Chebyshev polynomials Un(x), that by multiplying both sides of equation (2.3) by ξn, |ξ| < 1 and by summing up over n, it follows that: Z µ ¶ X+∞ +∞ X+∞ (tξ)n 1 ξnU (x) = e−t H 2x, − dt. (3.1) n n! n t n=0 0 n=0

By recalling the generating function of the polynomials Hn(x, y) stated in the relation (1.6) and by integrating over t, we end up with:

X+∞ 1 ξnU (x) = . (3.2) n 1 − 2ξx + ξ2 n=0

11 We can now state the related generating for the ﬁrst kind Chebyshev polynomials Tn(x) and for the polynomials Wn(x), by using the results proved in the previous theorem. Corollary 1 – Let x, ξ ∈ R, such that |x| < 1, |ξ| < 1; the generating functions of the polynomials Tn(x) and Wn(x) read:

X+∞ x − ξ ξnT (x) = (3.3) n+1 1 − 2ξx + ξ2 n=0 and X+∞ 8(x − ξ) (n + 1)(n + 2ξnW (x) = . (3.4) n+1 (1 − 2ξx + ξ2)3 n=0 Proof – By multiplying both sides of the relation (2.8) by ξn and by summing up over n, we obtain: X+∞ X+∞ X+∞ n n n ξ Tn+1(x) = x ξ Un(x) − ξ Un−1(x) n=0 n=0 n=0 that is X+∞ x ξ ξnT (x) = − n+1 1 − 2ξx + ξ2 1 − 2ξx + ξ2 n=0 which gives the (3.3). In the same way, by multiplying both sides of the second relation stated in the (2.7) by ξn and by summing up over n, we get

X+∞ X+∞ X+∞ n ξnU (x) = x ξnW (x) − ξnW (x) n+1 n n + 1 n−1 n=0 n=0 n=0 and then the thesis. These results allows us to note that the use of integral representations relat- ing Chebyshev and Hermite polynomials is a fairly important tool of analysis allowing the derivation of a wealth of relations between ﬁrst and second kind

Chebyshev polynomials and the Chebyshev-like polynomials Wn(x).

References

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12 [2] P. Appell – J. Kampéde Fériet, Fonctions Hypergéométriqueset Hyper- sphériques.Polynoˆmes d’Hermite, Gauthier-Villars, Paris, 1926.

[3] H.W. Gould, A.T. Hopper, Operational formulas connected with two ener- alizations of Hermite polynomials, Duke Math. J., 29, 1962, 51–62.

[4] P. Davis, Interpolation and Approximation, Dover, New York, 1975.

[5] L. C. Andrews, Special Functions for Engineers and Applied Mathematics, MacMillan, New York, 1958.

[6] G. Dattoli, C. Cesarano and D. Saccheti, A note on Chebyshev polynomials, Ann. Univ. Ferrara, 7 (2001), no. 47, 107–115.