Chebyshev polynomials of the second kind via raising operator preserving the orthogonality
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Chebyshev polynomials of the second kind via raising operator preserving the orthogonality Baghdadi Aloui1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract It is well known that monic orthogonal polynomial sequences {Tn }n≥0 and {Un }n≥0 , the Chebyshev polynomials of the first and second kind, satisfy the relation DTn+1 = (n + 1)Un (n ≥ 0). One can also easily check that the following “inverse” of the mentioned formula holds: U−1 (Un ) = (n + 1)Tn+1 (n ≥ 0), where Uξ = x(x D + I) + ξ D with ξ being an arbitrary nonzero parameter and I representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves Uξ which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence {a −n Un (ax)}n≥0 where a 2 = −ξ −1 , is actually the only monic orthogonal polynomial sequence which is Uξ classical (i.e. for which the application of the raising operator Uξ turns the original sequence into another orthogonal one). Keywords Orthogonal polynomials · Classical polynomials · Chebyshev polynomials · Second-order differential equation · Raising operator Mathematics Subject Classification Primary 33C45 · Secondary 42C05
1 Introduction An orthogonal polynomial sequence { pn }n≥0 is called classical, if { pn }n≥0 is also orthogonal. This characterization is essentially the Hahn-Sonine characterization (see [8,14]) of the classical orthogonal polynomials. In [9] Hahn gave similar characterization theorems for orthogonal polynomials pn such that the polynomials pn or Dq pn (n ≥ 1) are again orthogonal. Here is the difference operator and Dq is the q-difference operator given, x)− f (x) , q = 1. Notice that respectively, by f (x) = f (x + 1) − f (x) and Dq f (x) = f (q(q−1)x
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Baghdadi Aloui [email protected] Faculté des Sciences de Gabès, Département de Mathématiques, Cité Erriadh, 6072 Gabès, Tunisie
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B. Aloui
the differentiation, the difference and the q-difference are a lowering operators as they reduce the degree of any polynomial by exactly one. In the same context, we can cite the work [2] where we introduce the third-order differential operator Oc;1,3,2 . Here, a family of orthogonal polynomials pn (x) is in the Oc;1,3,2 -Hahn class if the polynomials (Oc;1,3,2 pn )(x) are again orthogonal. In a more general setting, let O be a linear operator acting on the space of polynomials which sends polynomials of degree n to polynomials of degree n + n 0 , where n 0 is a fixed integer (n ≥ 0 if n 0 ≥ 0 and n ≥ n 0 if n 0 < 0). We call a sequence { pn }n≥0 of orthogonal polynomials O-classical if {O pn }n≥0 is also orthogonal. In this paper, we consider the raising operator (raises the degree of a polynomial exactly by one), Uξ := x(x D + I) + ξ D where ξ is a nonzero free parameter and I represents the identity operator. We describe all the Uξ -classical orthogonal polynomial sequences. The basic idea has been deduced by starting from the transfer operat
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