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Asymptotic Properties of Biorthogonal Polynomials Systems Related to Hermite and Laguerre Polynomials

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Asymptotic Properties of Biorthogonal Polynomials Systems Related to Hermite and Laguerre Polynomials Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials Yan Xu School of Mathematics and Quantitative Economics, Center for Econometric analysis and Forecasting, Dongbei University of Finance and Economics, Liaoning, 116025, PR China Abstract In this paper, the structures to a family of biorthogonal polynomials that ap- proximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the gen- eralized Buchholz, Laguerre, Ultraspherical (Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from the theorems di- rectly. The relationship between Bernoulli and Euler polynomials are shown as a special case of the characterization theorem of the Appell sequence generated by α scaling functions. Keywords: Hermite Polynomial, Laguerre Polynomial, Appell sequence, Askey Scheme, B-splines, Bernoulli Polynomial, Euler polynomials. 2010 MSC: 42C05, 33C45, 41A15, 11B68 arXiv:1503.05387v1 [math.CA] 3 Feb 2015 ✩This work is supported by the Natural Science Foundation of China (11301060), China Postdoctoral Science Foundation (2013M541234, 2014T70258) and Outstanding Scientific In- novation Talents Program of DUFE (DUFE2014R20). ∗Email: yan [email protected] Preprint submitted to Constructive Approximation June 2, 2021 1. Introduction The Hermite polynomials follow from the generating function 2 xz z ∞ Hm(x) m e − 2 = z , z C, x R (1.1) m! ∈ ∈ m=0 X which gives the Cauchy-type integral 2 m! xz z (m+1) H (x)= e − 2 z− dz. (1.2) m 2iπ I 2 The derivatives of the Gaussian function, G(x)= 1 e x /2, produce the Her- √2π − m (m) 5 mite polynomials by the relation, ( 1) G (x) = H (x)G(x),m = 0, 1,.... − m Therefore the orthonormal property of the Hermite polynomials, 1 ∞ H (x)H (x)G(x)dx = δ , m! m n m,n Z−∞ can be considered as a biorthogonal relation between the derivatives of the Gaussian function, ( 1)nG(n) : n = 0, 1,... and the Hermite polynomials, { − } Hm : m =0, 1,... { m! } 10 The Hermite polynomials have been extensively studied since the pioneer article of C. Hermite [6] in 1864. It has many interesting properties and appli- cations in several branches of mathematics, physical sciences and engineering. A rich source of orthogonal polynomials, for instance, Gegenbauer [18], La- guerre [19], Charlier[13], Jacobi [14], Meixner-Pollaczek, Meixner, Krawtchouk 15 and Hahn-type polynomials [5] have asymptotic approximations in terms of Hermite polynomials, which is known as the famouse Askey scheme[21, 25, 20]. The asymptotic relations among other hypergeometric orthogonal polynomials and its q-analogue can be found in [4, 25]. The asymptotic representations of other families of polynomials, such as the generalized Bernoulli, Euler, Bessel 20 and Buchholz polynomials are also considered[8, 15, 26]. In [27], S. L. Lee extended the biorthogonal properties between the deriva- tives of the Gaussian function and Hermite polynomials to a family of scaling 2 functions with compact support and a family of Appell sequences which ap- proximate to the Gaussian function and Hermite polynomials respectively. The 25 Appell polynomials are also called scaling biorthogonal polynomials which are eigenfunctions of a linear operator and the distributional derivatives of φ are the eigenfunctions of its adjoint corresponding to the same eigenvalues [28]. In particular, the Appell polynomials generated by the uniform B-spline are the classical Bernoulli polynomials which asymptotic approximate to the Hermite 30 polynomials by suitably normalized[27]. The main objectives of this paper is to extend these properties to a family of non-scaling functions that approximate to the generating functions and to con- struct a family of biorthogonal polynomials that approximate to the Hermite polynomials and Laguerre polynomials respectively. The asymptotic properties 35 between Hermite, Laguerre and other orthogonal polynomials which are known as Askey Scheme are derived from these theorems as simple cases. The rela- tionship between the Appell sequence polynomials and the α-scaling compact support functions are also considered. This paper is organized as follows: In section 2, we present the framework 40 of the biorthogonal polynomials system which related to Hermite polynomials and Gaussian function. The characterization theorem of the Appell sequence generated by the scaling functions are also shown in this section. In section 3, as the applications of senction 2, the asymptotic relations among Bernoulli polyno- mials, Euler polynomials and B-splines are studied. The new identical relation 45 between Bernoulli and Euler polynomials are shown as a special case. In sec- tion 4, we generalize the asymptotic relationship to a family of hypergeometric orthogonal polynomials related to Hermite polynomials. The asymptotic prop- erties of Generalized Buchholz polynomials and Ultraspherical (Gegenbauer) polynomials are considered as the applications. In section 5, we generalize the 50 biorthogonal systems to the Laguerre polynomials and derive several asymptotic properties in Askey scheme. 3 2. Biorthogonal polynomials approximate to Hermite polynomial Let C∞(R) denoted for the space of infinitely differentiable functions. If φ : C∞(R) R is a linear functional, we shall write φ, ν = φ(ν),ν C∞(R). → h i ∈ The linear functional φ is continuous if and only if there is a compact subset K of R, a constant C > 0 and an integer k 0 such that ≥ φ, ν C max sup ν(j)(x) . |h i| ≤ j k x K | | ≤ ∈ We denote the space of distributions with compact support by ′(R). Integrable E functions and measures with compact supports belong to ′(R). If f is a com- E 55 pactly supported integrable function then it is associated with the distribution, which we still denote by f, defined by f,ν := ν(x)f(x)dx, ν C∞(R). h i R ∈ Z If m is a compactly supported measure on R, then it is associated with the distribution, which we still denote by m, defined by m,ν := ν(x)dm(x), ν C∞(R). h i R ∈ Z (n) Any φ ′(R) has derivatives φ of any order n and they are defined by ∈E φ(n),ν = ( 1)n φ, ν(n) , n =0, 1,.... − D E D E Taking a compactly supported distribution φ ′(R), then for any integer ∈ E n 0, ≥ (n) ( )z n n ( )z n n φ ,e · = ( 1) φ, z e · = ( 1) z φ(iz), − − D E D E 60 where φ( ) denote the Fourier transform of φ(x). b · If φ(0) = 0, b 6 ( )z n (n) e · n b ( 1) φ , = z (2.1) * − φ(iz)+ in a neighborhood of 0. Since φ is compactly supported, φ is analytic. So we b can define a sequence of polynomials, P , by the generating function m b exz ∞ P (x) = m zm. (2.2) m! φ(iz) m=0 X b 4 It follows from (2.1) and (2.2) that for any integer n 0, ≥ ∞ P (x) zn = ( 1)nφ(n), m zm, (2.3) − m! m=0 X which gives the biorthogonal relation P (x) ( 1)nφ(n), m = δ . (2.4) − m! m,n Definition 2.1. A sequence of polynomials, P (x): m N , is an Appell { m ∈ } sequence if Pm(x) is a polynomial of degree m and Pm′ (x)= mPm 1. − Differentiating (2.2) with respect to x and equating coefficients of zm in the resulting equation gives Pm′ (x)= mPm 1(x), m =1, 2,..., (2.5) − which implies Pm(x) are Appell sequence of polynomials. Therefore the distri- bution φ generates an Appell sequence of polynomials by the generating function exz 65 . φ(iz) b We consider a family of sequences of biorthogonal polynomials, P : m = { N,m 0, 1,...,N =1.2 ... , that are generated by a sequence of functions, φ , which } N converges to the Gaussian function exz ∞ P (x) = N,m zm. (2.6) ˆ m! φN (iz) m=0 X Definition 2.2. Let φ˜N be the standardized form of φN , φ˜N (x)= σN φN (σN x + µN ), 2 where µN and σN are the mean and variance of φN . Define the standardized form of the biorthonormal polynomials, P˜ , of P : m =0, 1,... by N,m { N,m } ˜ m PN,m(x)= σN− PN,m(σN x + µN ). (2.7) Then the following biorthogonal relations for the standardized biorthogonal poly- nomials follow from (2.4): ( 1)nφ˜(n), P˜ = δ , m,n 0. − N N,m m,n ∀ ≥ D E 5 Further, the generating functions of P˜N,m are given by[27] exz ∞ P˜ (x) = N,m zm. (2.8) m! φ˜ (iz) m=0 N X Theorem 2.1. Let φ˜N (x)bsatisfy the following conditions: 70 (1) There exist constant r> 0 such that for any ε, there is a sufficient large N0, for any N>N0, it holds z2 φ˜ (iz) e 2 ε, z < r. (2.9) N − ≤ | | b (2) Let P˜ (x): m = 0, 1,... be the biorthogonal polynomials generated { N,m } by the functions, φ˜N , as in (2.8). Then for each m =0, 1,..., P˜N,m(x) converges locally uniformly to the Her- 75 mit polynomial Hm(x) as N goes to infinity. ˜ Proof. Since φN (0) = 1, we can choose a neighborhood U of the origin so that 2 1 z 1 φ˜(iz) and e 2 for all z U. Take a circle C inside U with center ≥ 2 b ≥ 2 ∈ atb 0 and radius r so that (2.9) is satisfied. Noting exz exz ∞ P˜ (x) H (x) = N,m − m zm.
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