arXiv:1503.05387v1 [math.CA] 3 Feb 2015 oainTlnsPormo UE(DUFE2014R20). DUFE of Program Talents 2014T7025 novation (2013M541234, Foundation Science Postdoctoral sequ Appell the of theorem by characterization the s of are case polynomials special Euler a and Bernoulli between relationship The th rectly. the from Ber derived are (Gegenbauer), polynomials Meixner-Pllaczekare Ultraspherical and Meixner Laguerre, Buchholz, representa asymptotic eralized the properties, po these orthogonal of hypergeometric applications of the scheme Askey the verify turn syste the in sever from derived between are polynomials relation combinatorial and asymptotic polynomials the d are Therefore, polynomials polynomia Laguerre respectively. Generalized biorthogonal and of Hermite family the a to to proximate structures the paper, this In Abstract 00MSC: polynomials. 2010 Euler Polynomial, Bernoulli B-splines, Scheme, Askey Keywords: rpitsbitdt osrcieApoiainJn ,2 2, June Approximation Constructive to submitted Preprint colo ahmtc n uniaieEoois etrf Center Economics, Quantitative and Mathematics of School oeatn,DnbiUiest fFnneadEconomics, and Finance of University Dongbei Forecasting, ytm eae oHrieadLger polynomials Laguerre and Hermite to related systems ✩ ∗ α mi:yan Email: hswr sspotdb h aua cec onaino C of Foundation Science Natural the by supported is work This polynomials biorthogonal of properties Asymptotic cln functions. scaling emt oyoil aurePlnma,Apl sequence, Appell Polynomial, Laguerre Polynomial, Hermite 20,3C5 11,11B68 41A15, 33C45, 42C05, [email protected] a Xu Yan )adOttnigSinicIn- Scientific Outstanding and 8) ioig 105 RChina PR 116025, Liaoning, rEooercaayi and analysis Econometric or ia(1000,China (11301060), hina in ftegen- the of tions negenerated ence yoil.As lynomials. lorthogonal al oli Euler, noulli, sta ap- that ls oesdi- eorems s which ms, onas hown iscussed 021 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. (2.10) − z2 m! φ˜ (iz) e 2 m=0 N X The coefficients of the Taylor series (2.10) are represented by the Cauchy’s b integral formula
2 xz z ˜ m! e (e 2 φN (iz)) P˜N,m(x) Hm(x)= − dz. z2 − 2πi C m+1 ˜ I z φN (biz)e 2 Noting φ˜ (x) satisfy the condition (1), which meansb r> 0, for a real number N ∃ 80 A> 0, there is a sufficient large N0, for any N >N0, it holds
z2 A ˜ 2 φN (iz) e , z < r. (2.11) − ≤ σN | |
b
6 Therefore
2 xz z e e 2 φ˜ (iz) m! | | − N P˜N,m(x) Hm(x) dz z2 − ≤ 2π C m+1 ˜ | | I r φN (izb) e 2
xRe(z) A m! e b σ N dz z2 ≤ 2π C m+1 ˜ | | I r φN (iz) e 2 rx 4(m!)e A b m ≤ σN r
Since σ as N , It follows that for each m, P˜ H (x) N → ∞ → ∞ N,m(x) → m uniformly on compact sets.
2.1. Appell sequence generated by scaling functions
85 The refinement equation
φn(x)= αφn(αx y)dmn(y), x R,n =1, 2,..., (2.12) R − ∈ Z where α > 1 and m is a sequence of probability measures with finite first { n} and second moments. Equivalently, (2.12) can be expressed in term of Fourier transforms in the frequency domain in the form
µ µ φ (µ)= m φ , µ R. (2.13) n n α n α ∈ b b Theorem 2.2. If two Appell sequenceb polynomials, Pm(x) and Qm(x) are gen- eralized by exz ∞ zm = P (x) , (2.14) m m! ψ(iz) m=0 X and b exz ∞ zm = Q (x) (2.15) m m! φ(iz) m=0 X respectively. Then φ(x) is a α-scaling compact supported function with mask b ψ(x) if and only if
m m m α− Pk (αx) Qm k (αx)= Qm(2x). (2.16) k − Xk=0
7 Proof. Since
xz m xz m e α ∞ m z e α ∞ m z = α− P (x) , = α− Q (x) , iz m m! iz m m! ψ( ) m=0 φ( ) m=0 α X α X then b b
2xz m m e α ∞ m z ∞ m z = α− P (x) α− Q (x) (2.17) iz iz m m! m m! ψ( )φ( ) j=0 m=0 ! α α X X m m b b ∞ m m z = α− Pk(x)Qm k(x) . (2.18) k − m! m=0 ! X kX=0
Suppose that the Appell sequence polynomials, Pm(x) and Qm(x), satisfy (2.16),
m m m α− Pk (αx) Qm k (αx)= Qm(2x). k − Xk=0 Then
m ∞ m ∞ m m m αx αx z z α− Pk Qm k = Qm(x) . (2.19) k 2 − 2 m! m! m=0 ! m=0 X Xk=0 X 90 Therefore
xz m m e ∞ m m αx αx z = α− Pk Qm k (2.20) iz iz k 2 − 2 m! ψ( )φ( ) m=0 ! α α X Xk=0 ∞ zm exz b b = Q (x) = . (2.21) m m! m=0 φ(iz) X We see that b iz iz ψ( )φ( )= φ(iz), α α
which imply φ(x) is scalingb functionb withb mask ψ(x). Suppose that φ(x) is a α-scaling compact supported function with mask ψ(x), satisfying the scaling equation (2.12). Then the Fourier transform φ is given in (2.13) shows that b iz iz φ φ = φ(iz). (2.22) α α b b b
8 It follows that
∞ zm e2xz e2xz Q (2x) = = m m! iz iz m=0 φ(iz) ψ( )φ( ) X α α ∞ m ∞ m m z m z = b α− bP (bαx) α− Q (αx) m m! m m! m=0 ! m=0 ! X m X ∞ m m m z = α− Pk (αx) Qm k (αx) . k − m! m=0 ! X kX=0 Therefore m m m α− Pk (αx) Qm k (αx)= Qm(2x). k − Xk=0
95 3. Generalized Bernoulli polynomials, Euler Polynomials and B-splines
B-splines with order N, which is denoted as B ( ), is defined by the induc- N · tion as 1 if x [0, 1), ∈ B1(x) = 0 otherwise and for N 1 ≥ BN = B1 BN 1, ∗ − where denotes the operation of convolution which is defined by ∗ + ∞ (f g)(t) := f(t y)g(y)dy, ∗ − Z−∞ for f and g in L2(R).
The Fourier transform of BN (x) is
iω N 1 e− B (ω)= − , ω R. N iω ∈ b BN (x) also satisfies the scaling function as follow:
N 1 N B (x)=2 B (2x j), (3.1) N 2N j N − j=0 X
9 1 n where the mask ψN (k) := 2n k . Equivalently, (3.1) can be expressed in term of Fourier transforms in the frequency domain in the form:
BN (ω) = ψN (ω/2)BN (ω/2),
iω N 1+e− 100 b b b where the Fourier transform of the mask is ψN (ω)= 2 . The asymptotic properties of B-splines have a long history going back to the b physicist Arnold Sommerfeld who showed that Gaussian function can be approx- imated by B-splines point-wise in 1904 [1]. In 1992, Unser and his colleagues [17] proved that the sequence of normalized and scaled B-splines tends to Gaussian p 105 function in L space as the order N increases. L. H. Y. Chen, T. N. T. Goodman and S. L. Lee[16] considered the convergence orders of scaling functions which asymptotic to normality. A result due to Ralph Brinks [24] generalized Unser’s result to the derivatives of the B-splines. Yan Xu and R. H. Wang [29] gave the convergence orders of the approximation processes and showed the asymptotic
110 relationship among B-splines, Eulerian numbers and Hermite polynomials.
Theorem 3.1. [29] Let be k N, for N > k +2, the sequence of the k-th ∈ (k) derivatives, BN , of the B-spline converges to the k-th derivative of the Gaussian function
k+1 N 2 N N 1 x2 1 B(k) x + = Dk exp + O , (3.2) 12 N 12 2 √2π − 2 N r ! and k+1 2 k N (k) N N ( 1) lim B x + = − Hk(x)G(x), (3.3) d 12 N 12 2 √2π →∞ ( r !) where the limit may be taken point-wise or in Lp(R),p [2, ). ∈ ∞ Generalized Bernoulli[2, 3, 10] and Euler polynomials[22, 23] of degree m, N N order N and complex argument z, denoted respectively by Bm (z) and Em (z) can be defined by their generating functions,
ωN eωz ∞ BN (z) = m ωn, ω < 2π, (3.4) (eω 1)N m! | | m=0 − X 2N eωz ∞ EN (z) = m ωn, ω < π. (3.5) (eω + 1)N m! | | m=0 X 10 115 In paper [27], S. L. Lee has proved that the Appell polynomials generated by the uniform B-spline of order N are the generalized Bernoulli polynomials of order N and when suitably normalized they converge to the Hermit polynomials as N . Since the B-splines approximate the Gaussian function[16, 17, 24, → ∞ 27, 29], they can also be used as a filter in place of the Gaussian filter for linear
120 scale-space[30].
Corollary 3.1. [27]
m 2 12 N N N lim Bm z + = Hm(z). N N 12 2 →∞ r !
Proof. Recall that the uniform B-spline, BN (x), of order N, is the scaling func- tion satisfying N 1 N B (x)=2 B (2x j), N 2N j N − j=0 X 1 n where the mask ψN (k) := 2n k . Equivalently, the scaling equation can be expressed in term of Fourier transforms in the frequency domain in the form:
BN (ω) = ψN (ω/2)BN (ω/2),
iω N b b b 1+e− where the Fourier transform of the mask is ψN (ω) = 2 . From the fourier transform of B we have N b eω 1 N B (iω)= − . (3.6) N ω b N By (2.2), (2.4) and the generating function of Bm (z), the generalized Bernoulli polynomials, BN (z): m =0, 1,... , are biorthogonal to the derivatives of the { m } (n) B-splines, BN , BN (z) ( 1)nB(n)(z), m = δ . (3.7) − N m! m,n The standardized B-splines,
N N N B˜ (x)= B x + , N 12 N 12 12 r converges uniformly to the Gaussian function, G(x), and an estimate of the rate of convergence is given in [16, 27, 29] .
11 By Theorem 2.1, we have
m 2 12 N N N lim Bm z + = Hm(z). N N 12 2 →∞ r !
125
It is well known that the binomial distributions converge to the normal distribution in the sense that
[xN ] x 1 N 1 t2/2 lim = e− dt, (3.8) N 2N k √ →∞ 2π kX=0 Z−∞ where xN = √Nx/2 + N/2. Let σN = √Nx/2 and µN = N/2, then the
standardized binomial distributions, ψ˜N (x) := σN ψN (σN x + µN ), converges 2 uniformly to the Gaussian function G(x)= 1 e x /2. √2π − The Appell polynomials generated by the binomial distributions are the
130 generalized Euler polynomials of order N and when suitably normalized they converge to the Hermit polynomials as N . → ∞ Corollary 3.2.
m 2 4 N √N N lim Em ( z + )= Hm(z). (3.9) N N 2 2 →∞ Proof. From the Fourier transform of ψN (x) we have
eω +1 N ψ (iω)= . (3.10) N 2 b 1 n The Appell polynomials generated by binomial distributions, ψn(k) := 2n k , N are the generalized Euler polynomials, Em (z),m = 0, 1,..., that are biorthog-
onal to the derivatives of ψN (z), EN (z) ( 1)nψ(n)(z), m = δ . (3.11) − N m! m,n ˜N By Theorem (2.1), the normalized generalized Euler polynomials Em (z) con- verge to the Hermit polynomials, H (x), as N , m → ∞ m 2 4 N √N N lim Em ( z + )= Hm(z). (3.12) N N 2 2 →∞
12 Corollary 3.3. Generalized Euler and Bernoulli polynomials satisfy
m N 1 m N N Bm (z)= Ek (z)Bm k(z). (3.13) 2m k − Xk=0
iω N iω N 1+e− 1 e− Proof. Let ψN (ω) = 2 and BN (ω) = −iω . By (3.4) (3.5), we can see that the Euler and Bernoulli polynomials are generated by ψ (iω) and b b N 135 B (iω) respectively. Recall that the uniform B-spline, B , of order N, is the N N b scaling function satisfying b
BN (ω) = ψN (ω/2)BN (ω/2),
b b b and the Fourier transform of BN is
iω N 1 e− B (ω)= − , ω R. N iω ∈ Therefore by Theoremb 2.2, it holds
m N 1 m N N Bm (z)= Ek (z)Bm k(z). (3.14) 2m k − Xk=0
4. Generalized Buchholz polynomials and Ultraspherical (Gegenbauer)
140 polynomials
Theorem 4.1. Let P˜ (x): m = 0, 1,... be the polynomials sequence { N,m } f˜ (x,z) P˜ (x)zm generated by N = ∞ N,m . There are constants r > 0 and ˜ m=0 m! φN (iz) b xz A A > 0, such that for allP sufficient large N, it holds f˜N (x, z) e , − ≤ σN 2 ˜ z A ˜ and φN (iz) e 2 , for z < r. Then for each m =0 , 1,..., PN,m (x) con- − ≤ σN | |
145 verges b locally uniformly to the Hermit polynomial Hm(x) as N goes to infinity.
xz Remark 4.1. When f˜N (x, z)= e , theorem (4.1) turns to theorem (2.1).
˜ 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 at ≥ 2 b ≥ 2 ∈
b
13 xz A 0 and radius r, so that for all sufficient large N, it holds f˜N (x, z) e − ≤ σN 2 ˜ z A 150 and φN (iz) e 2 , for z < r. − ≤ σN | |
The b coefficients of the Taylor series
f˜ (x, z) exz ∞ P˜ (x) H (x) N = N,m − m zm (4.1) − z2 m! φ˜(iz) e 2 m=0 X are represented by the Cauchy’s integral formula b m! 1 f˜ (x, z) exz P˜ (x) H (x) = N dz N,m m m+1 z2 − 2πi C z ˜ − 2 I φN (iz) e ! 2 z ˜ xz ˜ m! e 2 fN (x,b z) e φ(iz) = − dz. z2 2πi C m+1 ˜ I z φ(iz)e 2 b b Therefore, for any given real numbers A > 0, there is a sufficient large N0,
for any N>N0, it holds
2 z xz e 2 f˜ (x, z) e φ˜(iz) m! N − P˜N,m(x) Hm(x) dz z2 − ≤ 2π C m+1 ˜ | | I z φ(iz) eb2 | | 2 2 2 z xz z xz z xz 2 ˜ b 2 2 ˜ m! e fN (x, z) e e + e e e φ(iz) − − dz ≤ 2π z2 | | C rm+1 φ˜( iz) e 2 b I 2 2 Re z xz xRez z 2 ˜ b 2 ˜ m! e fN (x, z) e + e e φ(iz) − − dz ≤ 2π z2 | | C rm+1 φ˜( iz) e 2 b I 2 r xr 4(m!)A e 2 + e b
m ≤ σN r Since σ as N , It follows that for each m, P˜ H (x) N → ∞ → ∞ N,m(x) → m 155 uniformly on compact sets. b
Generalized Buchholz and Ultraspherical (Gegenbauer) polynomials of de- N gree m, order N and complex argument x, denoted respectively by Pm (x) and N Cm (x), can be defined by their generating functions, N ∞ x(cotz 1 )/2 sin z N m e − z = P (x)z , z < π z m | | m=0 X
14 and
∞ 2 N N m (1 2xz + z )− = C (x)z , 1 x 1, ω < 1. − m − ≤ ≤ | | n=0 X
160 Buchholz polynomials are used for the representation of the Whittaker functions as convergent series expansions of Besse functions [9]. They appear also in the convergent expansions of the Whittaker functions in ascending powers of their order and in the asymptotic expansions of the Whittaker functions in descending powers of their order [12]. Explicit formulas for obtaining these polynomials may
165 be found in [11]. There are well known limits[15]
m 2 3 N √ 1 lim Pm ( 2 3Nx)= Hm(x) N N − m! →∞ and
m x 1 2 N lim (2N)− Cm = Hm(x). N √2N m! →∞ These limits give insight in the location of the zeros for large values of the limit parameter, and the asymptotic relation with the Hermite polynomials if the parameter N become large and x is properly scaled.
170 Many methods are available to prove these and other limits[15]. We can get these asymptotic results from theorem 4.1 as simple cases.
Lemma 4.1. For any z < π, | | z 12 z2 lim sincN = exp . N 2 N − 2 →∞ r ! Proof. Set z 12 LN (z) := N ln sinc . (4.2) " 2r N !#
z 12 Then with zN = 2 N , it holds q z 12 z 12 ln sinc( 2 N ) ln [sinc(z )] L (z) := N ln sinc =3z2 =3z2 N . N h 2 q i 2 " 2r N !# 3z /N zN
15 (1) (2) 1 Since it holds sinc(0) = 1, sinc (0) = 0 and sinc (0) = , for the limN , − 3 →∞ 175 and hence z 0, we may apply L’ Hˆopital’s rule twice: For any z < π, we N → | | have
(1) (2) 2 sinc (zN ) 2 sinc (zN ) lim LN (z) = 3z lim =3z lim N N 2z sinc(z ) N (1) →∞ →∞ N N →∞ 2sinc(zN )+2zN sinc (zN ) sinc(2)(0) z2 = 3z2 = . 2 − 2
It follows: For any z < π, | | z 12 z2 lim sincN = exp . N 2 N − 2 →∞ r !
Corollary 4.1.
m 2 3 N √ 1 lim Pm ( 2 3Nx)= Hm(x). (4.3) N N − m! →∞
N Proof. Let σN = 12 then q N m √ ω ∞ 6√2xσ cot ω 2σN sin m N ω N √ ω √2σN σ− P ( 12√2xσ ) = e− 2σN − N m − N √ ω m=0 2 √2σ ! X N N √ ω √2σN sin ω − 6 2xσN cot ω √2σ ˜ − √2σN − ˜ N Let fN (ω, x) = e and φN (iω) = ω . By √2σN Taylor theorem, for any ω < π and sufficient large N, it holds | | b
ω √2σN ω 3 ln f˜ (ω, x)= 6√2xσ cot = 6√2xσ + O(σ− ) . N − N √ − ω − N − √ N 2σN ! 3 2σN Therefore, for N + , we have → ∞
2xω lim f˜N (ω,z)= e . N →∞ By lemma 4.1, we have
N ω − sin 2 ˜ √2σN ω lim φN (iω) = lim ω = e . N + N → ∞ →∞ √2σN ! b
16 Therefore By Theorem 4.1, it holds
m 2 6 N √ 1 m lim Pm ( 2 6Nx)= Hm(√2x)√2 , (4.4) N N − m! →∞ equivalently, m 2 3 N √ 1 lim Pm ( 2 3Nx)= Hm(x). N N − m! →∞
Corollary 4.2.
m x 1 2 N lim (2N)− Cm = Hm(x). (4.5) N √2N m! →∞ N Proof. By the generating function of Cm (x),
∞ 2 N N m (1 2xz + z )− = C (x)z , − m n=0 X 180 it holds N ∞ 2 m N x m xz z − (2N)− 2 C z = 1 + m √ − √ 2N n=0 2N N X 2N (xz z2/2) 2xz z2 − 2xz z2 − = 1 − − − 2N .
2N 2xz z2 − 2xz z2 Let gN = 1 ( − ) − , then limN gN = e. Therefore − 2N →∞ h i 2 ∞ m N x m xz z /2 (2N)− 2 C z = g g− . m √ N N n=0 2N X and
xz xz lim gN = e N →∞ 2 z /2 z2/2 lim gN− = e− . N →∞ By Theorem (4.1), we have
m x 1 2 N lim (2N)− Cm = Hm(x). N √2N m! →∞
17 185 5. Biorthogonal systems relate to Generalized Laguerre polynomials
Laguerre polynomials, Ln(x), are solutions to the Laguerre differential equa- tion
xy′′ + (1 x)y′ + ny =0, n 0. − ≥
Laguerre polynomials is a class of orthogonal polynomials with weighting func- x tion w(x) = e− . The Rodrigues representation for the Laguerre polynomials
190 is
x n e d x n L (x)= (e− x ) (5.1) n n! dxn
and the generating function for Laguerre polynomials is
zx ∞ 1 − m (1 z)− e 1 z = L (x)z , z 1. (5.2) − − m | |≤ m=0 X (α) The generalized Laguerre polynomials, Lm , are also a class of orthogonal α x polynomials with weighting function w(x)= x e− and generated by
zx ∞ α 1 − (α) m (1 z)− − e 1 z = L (x)z , z 1. (5.3) − − m | |≤ m=0 X The Rodrigues representation for the generalized Laguerre polynomials is
α x n (α) x− e d x n+α L (x)= (e− x ). (5.4) n n! dxn
(0) 195 When α = 0, we have Ln (x)= Ln(x). (α) The explicit formula for Ln (x) is[7]
(α + 1) n ( n) xj L(α)(x)= n − j , (5.5) n n! (α + 1) j! j=0 j X where (α) = n (α + i 1), (α) = 1, for n = 1, 2, 3 .... When α = 0, the n i=1 − 0 (α) explicit formulaQ for generalized Laguerre polynomials, Ln (x), become
n n xk L (x)= ( 1)k . (5.6) n − k k! kX=0 By simply computing, we have
18 Proposition 5.1.
Ln′ (x)= Ln′ 1(x) Ln 1(x). (5.7) − − −
200 The orthogonality relation for the Laguerre polynomials is contained in
∞ (α) (α) α x Γ(α + n + 1) L (x)L (x)x e− dx = δ , α> 1, (5.8) m n n! mn − Z0 which can be considered as a biorthogonal relation between the derivatives of n α+n x d x e− (α) n and Laugerre polynomials Lm : m =0, 1,... , { dx Γ(α+n+1) } { }
dn xα+ne x L(α), − = δ . m dxn Γ(α + n + 1) m,n Corollary 5.1.
m m/2 (N) √ 1 lim ( 1) (2N)− Lm (x 2N + N)= Hm(x). N m! →∞ − Proof.
N 1 √Nz ∞ √2 zx m m/2 (N) m z − − ( 1) (2N)− L (x√2N + N)z = 1+ e 1+z/√2N e 1+z/√2N . − m √ m=0 2N X √Nz zx N+1 − √2 f˜ (x, z)= e 1+z/√2N , φ˜ (iz)= 1+ z e 1+z/√2N , then for N , we N N √N → ∞ have b
zx limN f˜N (x, z)= e , →∞ ˜ z2 limN φN (iz)= e 2 . →∞ b 205 By Theorem4.1, we have
m m/2 (N) 1 lim ( 1) (2N)− Lm (x√2N + N)= Hm(x). N − m! →∞
In this section, we consider a family of biorthogonal polynomials, P (x, α, ω): { m m =0, 1,... , generated by a sequence of functions, φ(x, α, ω), which converges } (α) to the generalized Laguerre polynomials, Lm (x).
19 Taking a compactly supported distribution φ ′(R), Let φˆ denote the ∈ E Laplace transform of φ. Then for any integer n 0, ≥ zx zx (n) n − n − n z φ , (1 z) e 1 z = φ(x),z e 1 z = z φ( ) (5.9) − − − 1 z D E D E − If φ(0) = 0, b 6 zx (1 z)ne −1 z φ(n), − − = zn (5.10) b z * φ( 1 z ) + − 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 zx (1 z)ne −1 z ∞ − − = P (x)zm. (5.11) z m φ( 1 z ) m=0 − X It follows from (5.10) and (5.11) that for any integer n 0, b ≥ ∞ n (n) m z = φ , Pm(x) z , (5.12) m=0 X D E which gives the biorthogonal relation
(n) φ , Pm(x) = δm,n. (5.13) D E m 210 Differentiating (5.11) with respect to x and equating coefficients of z in the resulting equation gives
Pm′ (x)= Pm′ 1(x) Pm 1(x), m =1, 2,..., (5.14) − − − which is similar to the property of Laguerre polynomials (Proposition 5.1),
Lm′ (x)= Lm′ 1(x) Lm 1(x). − − − α+n x x e− ˆ 1 If φ(x, α)= Γ(α+n+1) , the Laplace transform of φ(x, α) is φ(z, α)= (z+1)α+n+1 , ˆ z α+n+1 which implies φ( 1 z , α)=(1 z) . We have − − z n e− 1 z (1 z) φ(n)(x, α), − − = zn. (5.15) ˆ z * φ( 1 z , α) + − (α) in a neighborhood of 0. We can define a sequence of polynomials, Lm (x), by
215 the generating function
z 1 z n e− − (1 z) z ∞ α 1 1 z (α) m − = (1 z)− − e− − = L (x)z . ˆ z − m φ( 1 z , α) m=0 − X
20 α+n x x e− So the biorthogonal systems generated by function φ(x, α)= Γ(α+n+1) are La- guerre polynomials.
Theorem 5.1. Let φ(x, α, ω) satisfy the following conditions: (1) There are constants 0
z n e− 1 z (1 z) ∞ − − = P (x, α, ω)zm. ˆ z m φ( 1 z ,α,ω) m=0 − X Then for each m = 0, 1,..., P (x, α, ω): m = 0, 1,... converges locally { m } (α) 225 uniformly to the generalized Laguerre polynomial, Lm (x), as ω goes to c.
Proof. Since φ(0,α,ω) = 1, we can choose a neighborhood U of the origin so z 1 α+1 1 that φ( 1 z ,α,ω) 2 and (1 z) 2 for all z U. Take a circle C − b ≥ − ≥ ∈
completely contained in U, with centra at the origin 0 and radius r, so that b (5.16) is satisfied. The coefficients of the Taylor series
zx zx (1 z)ne −1 z e −1 z ∞ − − − = P (x, α, ω) L(α)(x) zm (5.17) z − (1 z)α+1 m − m φ( 1 z ,α,ω) m=0 − − X are represented by the Cauchy’s integral formula: b α+n+1 z 1 zx (1 z) φ( ,α,ω) (α) −1 z 1 z Pm(x, α, ω) L (x)= e − − − − dz − m 2πi m+1 z α+1 C z φ( 1 z ,α,ω)(1 z) I − b − (1b z)α+n+1 φ( z ,α,ω) 1 zx 1 z (α) −1 z − − − Pm(x, α, ω) Lm (x) e − dz − ≤ 2π C m+1 z α+1 I r φ( 1 z ,α,ω)b (1 z) − | − | z xRe ( 1− z ) 1 e − ε b dz ≤ 2π C m+1 z α+1 I r φ( 1 z ,α,ω) (1 z) − | − | z xRe( 1− z ) 4e − ε. ≤ b
21 It follows that for each m, Pm(x, α, ω) converges locally uniformly to the gen- (α) eralized Laguerre polynomial, Lm (x), as ω goes to c.
For the Meixner-Pollaczek polynomials, we have the generating function:
iω λ+ix iω λ ix ∞ (λ) m F (x, z)= 1 e z − 1 e− z − − = P (x; ω)z . (5.18) − − m n=0 X Corollary 5.2. The Laguerre polynomials can be obtained from Meixner-Pllaczek 1 1 1 230 polynomials by the substitution λ = (α + 1), x ω− x and letting ω 0. 2 →− 2 → α+1 2 x (α) lim Pn (− ; ω)= Ln (x). ω 0 2ω → Proof.
∞ ( α+1 ) x P 2 − ; ω zm n 2ω m=0 X ix α+1 iω − 2ω iω iω 2 1 ze = 1 ze 1 ze− − − − − 1 ze iω − −
α+1 z iω iω 2 n Let φ( 1 z ,α,ω)= 1 ze 1 ze− (1 z) , it holds − − − − α+1 z iω iω 2 n b lim φ( ,α,ω) = lim 1 ze 1 ze− (1 z) ω 0 1 z ω 0 → → − − − − α+1 b = lim 1 2z cos ω + z2 − 2 (1 z)n ω 0 → − − = (1 z)α+n+1. −
Since ix ix iω 2ω iω iω 2ω 1 ze − e− e − lim − = lim 1+ z − ω 0 1 ze iω ω 0 1 ze iω → − − → − − 1 e iω zx sin ω − − iω iω 2iz sin ω (1 ze iω)ω e− e − · − − = lim 1+ z − ω 0 iω → 1 ze− zx − 1 z = e − ,
by theorem (5.1), we have
α+1 2 x (α) lim Pn (− ; ω)= Ln (x). ω 0 2ω →
22 235 The generating function for Meixner polynomials Mn(x; β,c) is
x z β x ∞ (β)n n 1 (1 z)− − = M (x; β,c)z . − c − n! n n=0 X Corollary 5.3. The Laguerre polynomials can be obtained from Meixner poly- cx nomials by the substitution β = α +1, x 1 c and letting c 1. → − → (α) cx Ln (x) lim Mn ; α +1,c = . c 1 1 c (α) → − Ln (0) Proof.
∞ (α + 1) cx n M ; α +1,c zn n! n 1 c n=0 − X cx z 1 c cx − α 1 1 c = 1 (1 z)− − − − − c − cx z 1 c 1 c − α 1 = − (1 z)− − 1 z − − cx 1 c z(c 1) − α 1 = 1+ − (1 z)− − c(1 z) − − (1 z)c zx − z(c 1) −1 z z(c 1) − − α 1 = 1+ − (1 z)− − c(1 z) − −
Since (1 z)c zx − z(c 1) −1 z z(c 1) − − z 1 z x lim 1+ − = e− − (5.19) c 1 c(1 z) → − Therefore, by theorem(5.1), it holds
cx (α) lim(α + 1)nMn ; α +1,c = Ln (x). c 1 1 c → − Since n (α + 1) ( n) xj L(α)(x)= n − j , n n! (α + 1) j! j=0 j X α 240 then Ln(0) = (α + 1)n. Therefore
(α) cx Ln (x) lim Mn ; α +1,c = . c 1 1 c (α) → − Ln (0)
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