A Probabilistic Counterpart of Askey Scheme for Continuous Polynomials

(), 1–27 /..

A probabilistic counterpart of Askey scheme for continuous polynomials

Jean-Renaud Pycke

Abstract. Random variables corresponding to weight functions of the Askey scheme for continuous polynomials are introduced. Beside normal, Beta and Gamma distributions four new families of real random variables corresponding to Wilson, continuous dual Hahn, continuous Hahn and Meixner-Pollaczek polynomials are discussed. Formulas for the moments and asymptotic behavior of the tails of the distributions are provided. Keywords. Askey scheme, orthogonal polynomials, parametric model. 2010 Mathematics Subject Classiﬁcation. 42C05, 60E05.

1 Introduction

The main goal of this paper is to introduce families of real random variables de- rived in a natural way from the orthogonal families of the Askey-scheme for continuous polynomials. Both graphs (1.9) and (1.10) can be seen as probabilistic counterparts of the Askey scheme for continuous polynomials (1.1). The Askey scheme for continuous and discrete polynomials represents an out- standing achievement in the theory of special functions over the last decades. The Askey-scheme for continuous polynomials encompasses the theory of classical orthogonal polynomials of Laguerre, Hermite and Jacobi. It provides relations between these celebrated sequences of polynomials and includes them in a much wider class. The latter is much less familiar to non specialists and contains Wat- son, continuous dual Hahn, continuous Hahn, Meixner-Pollaczek polynomials. Whereas the connection between classical polynomials and parametric families of random variables as Gamma, Beta or Gaussian distributions are well-known, the random variables corresponding to this wider family have not been studied in a systematic way, even in a comprehensive survey of various relationships between orthogonal polynomials and probability theory as [8]. Excepting the Meixner- Pollaczek distribution, see e.g. [7] p.154, the use of these distributions as models remains a widely open field. Therefore the aim of the present paper is to con- tribute to filling this gap and provide basic results for further use of the fruitful Askey scheme in the field of probability and statistics. 2 J.-R. Pycke

Concerning the Askey scheme, notations, deﬁnitions and results about continuous polynomials of the Askey scheme we deal with in the present paper, complete and self-contained references for all formulas are [3], [4]. See also Chapter 1 and Appendix E in [8] and [5]. The Askey scheme of continuous polynomials admits the graph representation

Wilson (1.1)

r w Continuous dual Hahn Continuous Hahn

t ' Meixner-Pollaczek Jacobi

) w Laguerre

$ ~ Hermite

Each vertex corresponds to a parametric family of sequences of orthogonal polynomials named after the mathematician who introduced or studied the corresponding family. The arrows represent mathematical relations that will be made explicit below. With polynomial notations the graph becomes

2 Wk(x ; a, b, c, d)

s w 2 (a,b) Sk(x ; a, b, c) Hk (x)

x ' P (λ)(x; φ) P (α,β)(x)

& w L(α)(x)

Hk(x) Probabilistic Askey scheme 3

Arrows correspond to the limit relations

2 Wk(x ; a, b, c, d) 2 lim = Sk(x ; a, b, c) d→∞ (a + d)k 2 Wk((x + t) ; a − it, b − it, a + it, b + it) lim = Hk(x; a, b) t→∞ ((−2t)kk! 2 1−x α+1 α+1 β+1 β+1 Wk(t ; , , + it, − it) lim 2 2 2 2 2 = P (α,β)(x) t→∞ k!t2k S ((x − t)2; λ + it, λ − it, t cot φ) lim k = P (λ)(x; φ) t→∞ t k k!( sin φ )k Hλ+it,−t tan φ(x − t) lim k = P (λ)(x; φ) t→∞ k it k i ( cos φ )k α+1+it , β+1−it H 2 2 (− xt ) lim k 2 = P (α,β)(x) t→∞ (−t)k k

α+1 ( 2 ) x α lim Pk (− ; φ) = Lk (x) φ→0 2φ √ x λ − λ cos φ H (x) lim λ−k/2P (λ)( ; φ) = k λ→∞ k sin φ k! 2x lim P (α,β)(1 − ) = L(α)(x) β→∞ k β k x H (x) lim α−k/2P (α,α)(√ ) = k α→∞ k α 2kk! k (α) √ (−1) lim L (x 2α + α) = Hk(x). α→∞ k k!

Each family of orthogonal polynomials is related to a real weight-function via orthogonality relations (2.1). In view of applications in probability theory and statistics we will limit ourselves to parameters for which the weight function is non-negative. In that case after normalization the weight-function gives rise to a probability density function (p.d.f). The weight functions of the Askey-Wilson scheme and hypotheses on coefﬁcients are the following, the name of the corre- 4 J.-R. Pycke sponding orthogonal families being given each case.

2 1 Γ(a + ix)Γ(b + ix)Γ(c + ix)Γ(d + ix) fa,b,c,d(x) = (Wilson) 2π Γ(2ix) x > 0, R(a, b, c, d) > 0, non-real parameters occuring in conjugate pairs; (1.2) 2 1 Γ(a + ix)Γ(b + ix)Γ(c + ix) fa,b,c(x) = (continuous dual Hahn) 2π Γ(2ix) x > 0, R(a, b, c) > 0, non-real parameters occurring in a conjugate pair; (1.3) −1 2 fa,b(x) = (2π) |Γ(a + ix)Γ(b + ix)| , (continuous Hahn) x ∈ R, R(a, b) > 0; (1.4) f (λ)(x; φ) = (2π)−1e(2φ−π)x|Γ(λ + ix)|2 (Meixner-Pollaczek) x ∈ R, λ > 0, 0 < φ < π; (1.5) f (α,β)(x) = (1 − x)α(1 + x)β, −1 < x < 1, α, β > −1 (Jacobi); (1.6) f (α)(x) = xαe−x, x > 0, α > −1 (Laguerre) ; (1.7)

−x2 f0(x) = e , x ∈ R (Hermite). (1.8)

Remark 1.1. Hypotheses upon (a, b) in (1.2), (1.3), (1.4) are nothing but the same as those appearing in the deﬁnition the corresponding polynomials. Note that in (1.4) restriction in terms of conjugation as in (1.2) or (1.3) disappeared, a and b arising as limits of two pairs of conjugate coefﬁcients, see relation (1.12).

The p.d.f. corresponding to the weight-function f will be denoted by f˜. Note that whereas the weight function corresponding to a given family is subject to variations in the literature, up to a constant factor, the corresponding p.d.f. is unique. Probabilistic Askey scheme 5

Our Askey scheme for p.d.f.’s will therefore be the following.

f˜a,b,c,d(x) (1.9)

t y f˜a,b,c(x) f˜a,b(x)

y % f˜(λ)(x; φ) f˜(α,β)(x)

% y f˜(α)(x)

Ó f˜0(x)

In Proposition 2.1 relation (2.2) gives normalization constants as a function of one of the coefﬁcient related to the corresponding orthogonal family. It enables us to give the explicit closed form of the p.d.f.’s for the seven families, see formulas (3.3),(4.4), (5.3),(6.2),(7.3), (8.3) and (9.3). Furthermore formulas (3.4), (4.5), (5.4), (6.2), (7.4), (8.4), (9.5) describe when necessary the asymptotic behavior of distribution tails providing as a corollary the domain of the moment generating function (m.g.f.). Concerning the latter, we will not give more detail as a closed form when it is available and the limit relations. We postpone this issue to a forthcoming paper. With obvious notations random variables with these distributions will be denoted by

Wil2(a, b, c, d), CDHahn2(a, b, c), CHahn(a, b, ), MeixPol(λ, φ), Jac(α, β), Lag(α), Her.

Remark 1.2. Wilson and continuous dual Hahn polynomials are deﬁned as polynomials of the variable x2. This is the reason why we consider the corresponding p.d.f as that of squared random variables.

Remark 1.3. As mentioned above the random variables related to Jacobi, Laguerre and Hermite polynomials are closely related to the well-known Beta, Gamma and 6 J.-R. Pycke

Gaussian densities respectively. More precisely, if these random variables and their densities are Γ(α + β) Beta(α, β) : xα(1 − x)β (0 < x < 1) Γ(α)Γ(β) βα Gamma(α, β) : xα−1e−βx (x > 0) Γ(α) 2 2 −1/2 − x N(µ, σ ) : (2π) e 2σ2 (x ∈ R) it is readily checked that 1 − Jac(α − 1, β − 1) Lag(α − 1) Beta(α, β) = , Gamma(α, β) = , 2 β √ N(µ, σ2) = µ + σ 2 Her.

The Askey scheme for these random variables is the following.

Wil2(a, b, c, d) (1.10)

s w CDHahn2(a, b, c) CHahn(a, b)

v ' MeixPol(λ, φ) Jac(α, β)

( v Lag(α)

! ~ Her

It is shown in Proposition 2.2 that the sequence of moments of these random variables is related to the sequence of polynomials via the elementary but far-reaching formula (2.3). In view of applications we give the ﬁrst two or three polynomials of each family. They allow to compute the expectation and variance of our random variables, see formulas (3.2), (4.3), (5.2), (6.1), (7.2), (8.2), (9.4). The limit relations between p.d.f.’s corresponding to limit relations between polynomials are given by (3.5), (3.7), (3.8), (4.6), (5.5). (5.6), (6.3), (6.4), (7.5), (7.6) and (8.5). These convergences of densities, when combined with Lemma Probabilistic Askey scheme 7

2.3 which is a particular case of Scheffé’s Lemma, will imply in turn the following limit relations between our random variables.

lim Wil2(a, b, c, d) = CDHahn2(a, b, c) d→∞ (1.11) lim Wil2(a − it, b − it, a + it, b + it) − t = CHahn(a, b) t→∞ (1.12) 2 α + 1 α + 1 β + 1 β + 1 lim 1 − Wil4( , , + it, − it) = Jac(α, β) t→∞ t2 2 2 2 2 (1.13) lim t − CDHahn2(λ + it, λ − it, t cot φ) = MeixPol(λ, φ) t→∞ (1.14) lim t + CHahn(λ + it, −t tan φ0) = MeixPol(λ, φ) t→∞,tan φ>0 (1.15) 2 α + 1 + it β + 1 + it lim − CHahn( , ) = Jac(α, β) t→∞ t 2 2 (1.16) α + 1 lim −2φMeixPol( , 2φ) = Lag(α) φ→0+ 2 (1.17) sin φ √lim √ (MeixPol(λ, φ) + λ cot φ) = Her (λ,cos φ λ)→(∞,0) λ (1.18) β lim (1 − Jac(α, β)) = Lag(α) β→∞ 2 (1.19) √ lim αJac(α, α) = Her α→∞ (1.20) Lag(α) − α lim √ = Her α→∞ 2α (1.21) 8 J.-R. Pycke

2 Orthogonal polynomials and probability measures

In this Section we recall basic facts, results and references concerning sequences of orthogonal polynomials. For further details and proofs concerning this topic see Chapters 5, 6 in [2], Chapter V in [6] and Chapter 1 in [8]. Throughout this paper x, possibly with a subscript index, will always denote a real variable. The real line is endowed with the standard Lebesgue measure. A sequence of real polynomials {Pk(x), k ∈ N} where Pk is of degree k is said to be an orthogonal system with respect to the weight function f with support S ⊆ R if it satisﬁes orthogonality relations Z 2 Pk(x)P`(x)f(x)dx = dkδk`. (2.1) S

If dk = 1 for each k, the system is said to be orthonormal. The following result is straightforward.

Proposition 2.1. With notations as above relations

Z 2 Z 2 P0 d0Pk(x) d0P`(x) P0 2 f(x)dx = 1, · · 2 f(x)dx = δkl S d0 S dkP0 d`P0 d0 hold. In other words the function

P 2 ˜ 0 f(x) = 2 f(x), x ∈ S (2.2) d0 is the probability density function associated with the orthonormal family

d0 pek(x) = Pk(x)(k ∈ N) dkP0

Let X be a random variable with density f˜ and k ∈ N. The k-th moment about the origin of X is denoted and deﬁned by

0 r µk = EX , r = 0, 1, 2, ...

0 0 So µ0 = 1 and µ1 = µ is the mean, or expected value of X. For each k ≥ 1 the orthogonality relation between Pk and the constant polynomial P0 implies the equality Z f˜(x)Pk(x)dx = 0. S Therefore we can state the following result, providing a method for computing all moments of X by induction. Probabilistic Askey scheme 9

∗ Proposition 2.2. For each k ∈ N holds the equality

EPk(X) = 0. (2.3)

In other words if k X j Pk(x) = ajx j=0 then k X 0 ajµj = 0. j=0 From this Proposition the explicit expressions obtained for the ﬁrst values of k are the following. If

2 P0(x) = a0,P1(x) = a1x + b1,P2(x) = a2x + b2x + c2 then b1 2 0 c2 b1b2 EX = µ = − , EX = µ2 = − + (2.4) a1 a2 a1a2 and therefore the variance is given by

2 0 2 c2 b1b2 b1 VX = µ2 − µ = − + − 2 . (2.5) a2 a1a2 a2 The following Lemma is a particular case of the well-known Scheffé’s Lemma. It enables us to infer limit relations between random variables from relations between pdf’s.

Lemma 2.3. Assume −∞ ≤ t0 ≤ ∞ and let V0 denote a neighborhood of t0. 1 Assume that for each t ∈ V0, x ∈ R 7→ at(x) is a strictly monotone C real function and that ft is the probability density function of a random variable Xt. Let f denote the probability density function of a random variable X. If for almost each x ∈ R 0 lim |at(x)|f(at(x)) = f(x), t→t0 then the convergence in total variation

−1 lim at (Xt) −→ X t→t0 holds. 10 J.-R. Pycke

0 Proof. After noticing that |at(x)|f(at(x)) is the p.d.f. of the random variable −1 at (Xt), the result is a consequence of Scheffé’s lemma, as stated in §2.9, corollary 2.30 p.22 in [9]. In the particular case of the Askey-scheme we will need to use the following notations and classical results from the the theory of special functions. The hypergeometric series rFs is deﬁned by ! ∞ Qr k a1, ..., ar X i=1(ai)k z rFs z = 1 + Qs b , ..., b (bj)k k! 1 s k=1 j=1 where for (z, n) ∈ C × N, (z)n denotes the Pochhammer’s symbol deﬁned by n Y (z)0 = 1, (z)n = (z + i − 1) for n ≥ 1. i=1 Recall that if z = x + iy denotes a complex number with real part x = R(z), imaginary part y = I(z) and if a = |a| exp(i arg a) with arg a ∈ (−π, π] then one has |az| = |a|xe−y arg a. (2.6) Concerning Euler’s Γ–function, beside the fundamentl relation

Γ(z + 1) = zΓ(z)(z ∈ C) useful relations for x, y ∈ R, α, β, z ∈ C are Γ(2z) = (2π)−1/222z−1Γ(z)Γ(z + 1/2) (2.7) ( 2x sinh 2πx 4x2 as x → 0+, |Γ(2ix)|−2 = ∼ (2.8) π (x/π)e2πx as x → ∞, |Γ(x + iy)|2 ∼ |Γ(x)|2 as x → ∞ (2.9) |Γ(x + iy)|2 ∼ 2πe−π|y||y|2x−1 as |y| → ∞, (2.10) |Γ(z + ix)|2 ∼ 2πe−π|I(z)+x||x|2R(z)−1, as |x| → ∞ (2.11) Γ(z + α) ∼ zα−β as R(z) → ∞, (2.12) Γ(z + β) |Γ(z)|2 ∼ 2π|e−2zz2z−1| as R(z) → ∞ = 2πe−2R(z)|z|2R(z)−1e−2I(z) arg z (2.13) See, e.g., formulas 4.2.9, 13 and 6.1.25, 26, 29, 39, 45, 46 p. 256-257 in [1] or The- orem 1.4.1 p.18, last relation of §1.4 and Corollary 1.4.4 p.21-22 in [2], the last but one p.12 and ﬁrst p.13 formulas in [6]. Probabilistic Askey scheme 11

3 Wilson polynomials and distributions

About the weight function (1.2) and the associated Wilson polynomials see formulas (3.8.1), (3.8.3), Deﬁnition 3.8.1 in §3.8 of [2], [8] p.11, formulas (1.1.1) − (1.1.2) in [3]. For each k ∈ N, the k-th Wilson polynomial Wk(.; a, b, c, d) is deﬁned by

2 ! Wk(x ; a, b, c, d) −k, k + a + b + c + d − 1, a + ix, a − ix =4 F3 1 . (a + b)k(a + c)k(a + d)k a + b, a + c, a + d

Relation (1.2) suggests that Wilson polynomials should be real-valued and symmetric with respect to a, b, c and d as their weight function is. It can be seen directly via the generating functions

! ! 2 k a + ix, d + ix b − ix, c − ix X Wk(x ; a, b, c, d) t 2F1 t 2F1 t = a + d b + c (a + d)k(b + c)k k! k∈N ! ! 2 k a + ix, b + ix d − ix, c − ix X Wk(x ; a, b, c, d) t 2F1 t 2F1 t = a + b d + c (a + b)k(c + d)k k! k∈N ! ! 2 k a + ix, d + ix b − ix, c − ix X Wk(x ; a, b, c, d) t 2F1 t 2F1 t = a + d b + c (a + d)k(b + d)k k! k∈N

Wilson polynomials satisfy the orthogonality relations

Z ∞ 2 2 2 fa,b,c,d(x)Wk(x ; a, b, c, d)W`(x ; a, b, c, d)dx = dk(a, b, c, d)δk` 0 with (k + a + b + c + d − 1) k! d2 (a, b, c, d) = k c (a, b, c, d) k Γ(2k + a + b + c + d) k where ck(a, b, c, d) = Γ(k+a+b)Γ(k+a+c)Γ(k+a+d)Γ(k+b+c)Γ(k+b+d)Γ(k+c+d), whence

Γ(a + b)Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)Γ(c + d) d2(a, b, c, d) = . (3.1) 0 Γ(a + b + c + d) 12 J.-R. Pycke

The ﬁrst Wilson polynomials and moment of the correspond r.v. are

2 W0(x ; a, b, c, d) = 1, 2 2 W1(x ; a, b, c, d) = abc + abd + acd + bcd − (a + b + c + d)x , 2 4 W2(x ; a, b, c, d) = (a + b + c + d + 1)(a + b + c + d + 2)x X X X X X X −(2 a2bc + 4 ab + a2 + a + 2 a2b + 8 abc + 8abcd)x2 X X X X X X + (abc)2 + 2 (a2b2cd) + a2bc + a2b2c + 4 a2bcd + abc + 6abcd, abc + abd + acd + bcd Wil2(a, b, c, d) = E a + b + c + d (3.2) where the sums are computed over all permutations of a, b, c and d. In view of (1.2), (2.2) and (3.1) we obtain for the pdf of Wil2(a, b, c, d) the expression Γ(a + b + c + d) f˜ (x) = f (x) a,b,c,d Γ(a + b)Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)Γ(c + d) a,b,c,d (3.3) As a straightforward consequence of (2.8), (2.11), keeping in mind the hypothesis made upon a, b, c, d we obtain the asymptotic relations 2 ˜ 2|Γ(a)Γ(b)Γ(c)Γ(d)| 2 + fa,b,c,d(x) ∼ 2 x as x → 0 , πd0(a, b, c, d) 2 −2πx ˜ 8π e 2(a+b+c+d)−3 fa,b,c,d(x) ∼ 2 x as x → ∞, d0(a, b, c, d)  t < 2π, tWil2(a,b,c,d)  Ee < ∞ ⇐⇒ or  t = 2π, a + b + c + d < 1. (3.4)

Proposition 3.1 (From Wilson to Continuous dual Hahn). Assume a, b, c and d satisfy both conditions stated in (1.2) and (1.3), say a, d ∈ R and b = c. Then for each x > 0 the convergence

lim fã,b,c,d(x) = fã,b,c(x) (3.5) d→∞ holds where fã,b,c is the continuous dual Hahn pdf given by (4.4) holds. Conse- quently relation (1.11) is valid. Probabilistic Askey scheme 13

Proof. Note ﬁrst that

Γ(a + b + c + d)|Γ(d + ix)|2 f˜ (x) = f˜ (x). (3.6) a,b,c,d Γ(a + d)Γ(b + d)Γ(c + d) a,b,c

Then in view of (2.9), (2.12) we obtain„ as d → ∞,

Γ(a + b + c + d)|Γ(d + ix)|2 Γ(a + b + c + d)Γ(d)2 ∼ Γ(a + d)Γ(b + d)Γ(c + d) Γ(a + d)Γ(b + d)Γ(c + d) ∼ d(a+b+c)−(a+b+c) = 1 ensuring both desired convergences, with the help of Lemma 2.3 for the latter.

Proposition 3.2 (From Wilson to Continuous Hahn). Assume a, b satisfy conditions in (1.2). Then for each x ∈ R,

lim f˜ (x + t) = f˜a,b(x) (3.7) t→∞ a−it,b−it,a+it,b+it where f˜a,b is the continuous Hahn pdf given by (5.3). Consequently, (1.12) holds.

Proof. Note that

2 Γ(a + i[x + 2t])Γ(b + i[x + 2t]) f˜ (x + t) = f˜a,b(x) a−it,b−it,a+it,b+it Γ(a + b + 2it)Γ(2i[x + t])

Then by using (2.12) we obtain, as t → ∞,

a+ix+b+ix−(a+b−2ix) 2 ˜ ˜ fa−it,b−it,a+it,b+it(x + t) ∼ |(2it) | fa,b(x) = fa,b(x) and the (3.7) is proved. Lemma 2.3 enables us to conclude.

Proposition 3.3 (From Wilson to Jacobi). Assume α, β > −1. Then for each x ∈ (−∞, 1) r t ˜ 1 − x ˜(α,β) lim p f α+1 , α+1 , β+1 +it, β+1 −it(t ) = f (x). (3.8) t→∞ 2 2(1 − x) 2 2 2 2 2

Therefore relation (1.13) holds. 14 J.-R. Pycke

Proof. On the ﬁrst hand

r ˜ 1 − x Γ(α + β + 2) f α+1 , α+1 , β+1 +it, β+1 −it(t ) = α+β+2 2 2 2 2 2 4 2πΓ(α + 1)Γ(β + 1)|Γ( 2 + it)| q q q 2 Γ( α+1 + it 1−x )2Γ( β+1 + it[1 + 1−x ])Γ( β+1 + it[−1 + 1−x ]) 2 2 2 2 2 2 × q 1−x Γ(2it 2 ) (3.9)

On the other hand we have the asymptotic equivalences, as t → ∞, r r α + 1 1 − x 1 − x 1 − x |Γ( + it )|4 ∼ 4π2t2α( )α exp{−2πt } 2 2 2 2 r r r β + 1 1 − x 1 − x 1 − x |Γ( + it[1 + ])|2 ∼ 2πtβ[1 + ]β exp{−πt[1 + ]} 2 2 2 2 r r r β + 1 1 − x 1 − x 1 − x |Γ( + it[−1 + ])|2 ∼ 2πtβ|1 − |β exp{−πt|1 − |} 2 2 2 2 α + β + 2 |Γ( + it)|4 ∼ 4π2t2α+2β+2 exp{−2πt} 2 r r 1 − x 1 − x 1 − x |Γ(2it )|2 ∼ πt−1( )−1/2 exp{−2πt }. 2 2 2

In the asymptotic equivalence deduced in turn for (3.9) the argument of the expo- nential will be

r r ( q 1 − x 1 − x −2πt( 1−x − 2) < 0 if x < −1 −πt(1+ +|1− |−2) = 2 2 2 0 if −1 < x < −1 so that for x < −1 the expected convergence to 0 holds. If −1 < x < 1 above equivalences after simpliﬁcations imply r r ˜ 1 − x 4 1 − x ˜(α,β) f α+1 , α+1 , β+1 +it, β+1 −it(t ) ∼ f (x) 2 2 2 2 2 t 2 p which is equivalent to (3.8). Finally applying Lemma 2.3 with at(x) = t (1 − x)/2, 0 p −1 2 2 |at(x)| = t/(2 2(1 − x)) and at (y) = 1 − 2y /t we see that (1.13) is then straightforward consequence of (3.8). Probabilistic Askey scheme 15

4 Continuous dual Hahn distributions

About the weight function (1.3) and continuous dual-Hanhn polynomials see §1.3 in [3], relation (6.10.7) and the following remark p.332-333 in [2], [8] p.11. For each k ∈ N the k−th Continuous dual Hahn polynomials Sk(; a, b, c) is deﬁned by 2 ! Sk(x ; a, b, c) −k, a + ix, a − ix = 3F2 1 . (4.1) (a + b)k(a + c)k a + b, a + c

The invariance property of Sk under permutations of a, b and c appears clearly, as for Wilson polynomials, by considering the generating functions ! 2 k −c+ix a + ix, b + ix X Sk(x ; a, b, c) t (1 − t) 2F1 t = a + b (a + b)k k! k∈N ! 2 k −b+ix a + ix, c + ix X Sk(x ; a, b, c) t (1 − t) 2F1 t = a + c (b + c)k k! k∈N ! 2 k −a+ix b + ix, c + ix X Sk(x ; a, b, c) t (1 − t) 2F1 t = b + c (b + c)k k! k∈N Keeping this property in mind the real character of Sk then follows from the fact that if a ∈ R then b and c are real or conjugate and (4.1) enables to conclude. These polynomials satisfy the orthogonality relations Z ∞ 2 2 2 fa,b,c(x)Sk(x ; a, b, c)S`(x ; a, b, c)dx = dk(a, b, c)δk` 0 where 2 dk(a, b, c) = Γ(k + a + b)Γ(k + a + c)Γ(k + b + c)k!, whence 2 d0(a, b, c) = Γ(a + b)Γ(a + c)Γ(b + c). (4.2) From deﬁnition (4.1) one obtains for the ﬁrst continuous dual Hahn polynomials and moments of CDHahn2(a, b, c) the expressions 2 S0(x ; a, b, c) = 1, 2 2 S1(x ; a, b, c) = −x + ab + ac + bc, 2 4 X X 2 S2(x ; a, b, c) = x − (2 ab + 1 + 2 a)x X X X X + (ab)2 + 2 (a2bc) + ab + a2b + 4abc,

2 ECDHahn (a, b, c) = ab + ac + bc. (4.3) 16 J.-R. Pycke

From (1.3), (2.2) and (4.2) and we obtain for the p.d.f. of CDHahn2(a, b, c), f (x) f˜ (x) = a,b,c (x > 0) a,b,c Γ(a + b)Γ(a + c)Γ(b + c) Γ(a + ix)Γ(b + ix)Γ(c + ix) 1 = . (4.4) 2πΓ(a + b)Γ(a + c)Γ(b + c) |Γ(2ix)|2 By using (2.8), (2.11) and the hypothesis made upon a, b, c we see that the tails of this distribution and the domain of its m.g.f. satisfy

 2|Γ(a)Γ(b)Γ(c)|2x2  as x → 0+,  πΓ(a+b)Γ(a+c)Γ(b+c) f˜a,b,c(x) ∼  4πe−πxx2(a+b+c)−2  Γ(a+b)Γ(a+c)Γ(b+c) as x → ∞, ( tCDHahn2(a,b,c) t < π or Ee < ∞ ⇐⇒ (4.5) t = π and a + b + c < 1/2.

Proposition 4.1 (From continuous dual Hahn to Meixner-Pollaczek). For each x ∈ R, convergence (λ) lim f˜λ+it,λ−it,t cot φ(t − x) = f˜ (x; φ), (4.6) t→∞ where f (λ)(.; φ) is given by (6.2), holds, and consequently (1.15) is valid.

Proof. We obtain successively

5 Continuous Hahn distributions

For the weight function (1.4) and the associated continuous Hahn polynomials see §1.4, relation (6.10.10) p. 333 in [2] and [8] p. 11. Probabilistic Askey scheme 17

(a,b) For each k ∈ N the k-th continuous Hahn polynomial Hk (.) is deﬁned by ! (a,b) k (a + a)k(a + b)k −k, k − 1 + a + a + b + b, a + ix H (x) = i 3F2 1 . k k! a + a, a + b The fact that these polynomials are real-valued, symmetric with respect to a, b as their weight function is becomes clear in view of the formal generating function ! ! a + ix, b + ix a − ix, b − ix X (a,b) k 2F0 − it 2F0 it ∼ H (x)t . − − k k∈N They satisfy the orthogonality relations Z (a,b) (a,b) 2 2 fa,b(x)Hk (x)H` (x)dx = dk(a, b) δk` R where Γ(k + a + a)Γ(k + b + b)|Γ(k + a + b)|2 d2 (a, b) = , k (2k + a + a + b + b − 1)Γ(k + a + a + b + b − 1)k! whence Γ(a + a)Γ(b + b)|Γ(a + b)|2 d2(a, b) = . (5.1) 0 Γ(a + a + b + b) The ﬁrst continuous Hahn polynomials and moments of MeixPol(a, b) are

(a,b) H0 (x) = 1 (a,b) H1 (x) = 2I(ab) + 2xR(a + b) (a,b) 2 2 H2 (x) = {2(R(a + b)) + 3R(a + b) + 1}x +{4I(ab)R(a + b + 1) + I[a(a + 1) + b(b + 1)]}x +2I(a2)I(b2) + 2R(ab) − 2(R(a)R(b2) + R(b)R(a2)) +4I(ab)(R(ab) + R(a + b)), I(a, b) CHahn(a, b) = − . (5.2) E R(a + b) We see from (1.4), (2.2) and (5.1) that the continuous Hahn p.d.f. is given by

From (2.11) we obtain the asymptotic behavior

2πΓ(a + a + b + b) −π|2x+I(a+b)| 2R(a+b)−2 f˜a,b(x) ∼ e |x| as |x| → ∞. Γ(a + a)Γ(b + b)|Γ(a + b)|2 (5.4)

Proposition 5.1 (From continuous Hahn to Meixner-Pollaczek). Assume x ∈ R, φ 6= π/2. Then n (λ) lim fλ+it,t tan φ(x − t) = f (x; φ) (5.5) t→∞,t tan φ>0 and therefore convergence in law (1.15)

Proof. Note ﬁrst that

Since in both cases t tan φ > 0, we see from (2.6), (2.12) that

Γ(t tan φ + it − ix) | |2 ∼ |(t tan φ + it)−2λ−2ix| Γ(t tan φ + it + λ)

t −2λ 2x( π −φ) ∼ ( ) e 2 , cos φ Γ(2λ + 2t tan φ) ∼ (2t tan φ)2λ Γ(+2t tan φ) and the simple convergence is proved. Apply Lemme 2.3 to conclude.

Proposition 5.2 (From continuous Hahn to Jacobi). . For each x ∈ R,

( ˜ t ˜ xt fα,β(x) if |x| ≤ 1 lim f α+1+it , β+1−it (− ) = (5.6) t→∞ 2 2 2 2 0 0 if |x| > 1 and therefore the convergence in law (1.16) holds.

Proof. Note ﬁrst that

|x − 1| |x + 1| + − 1 = max(0, |x| − 1)(x ∈ ) 2 2 R Probabilistic Askey scheme 19

α+1+it(1−x) β+1−it(1+x) 2 xt Γ(α + β + 2)|Γ( 2 )Γ( 2 )| f α+1+it , β+1−it (− ) = α+β+2 2 2 2 2 2πΓ(α + 1)Γ(β + 1)|Γ( 2 + it)| Γ(α + β + 2)|2π exp{−π t|1−x| }( t|1−x|) )α · 2π exp{−π t|1+x| }( t|1+x| )β ∼ 2 2 2 2 2π · 2πΓ(α + 1)Γ(β + 1) exp{−πt}tα+β+1 2 Γ(α + β + 2)|1 − t|α|1 + t|β |x − 1| |x + 1| = exp{−πt( + − 1)} t 2α+β+1Γ(α + 1)Γ(β + 1) 2 2 and the claimed result as well as (1.16) readily follow.

6 Meixner-Pollaczek polynomials and distributions

About distribution (1.5) and the associated Meixner-Pollaczek polynomials see Exercice 37 p.348 in [2] §1.7 in [3] and [8] p.11 (up to a misprint for the density denoted by ρ in this reference). The Meixner-Pollaczek polynomials are deﬁned, for each k ∈ N, by ! (λ,φ) (2λ)k ikφ −k, λ + ix −2iφ P (x) = e 2F1 1 − e . k k! 2λ

Their real-valued character appears more clearly from the generating function

iφ −λ+ix −iφ −λ−ix X (λ,φ) k (1 − te ) (1 − te ) = Pk (x)t . k∈N They satisfy the orthogonality relations Z (λ,φ) (λ,φ) 2 Pk (x)P` (x)fλ,φ(x)dx = dk(λ, φ)δk` R with Γ(k + 2λ) Γ(2λ) d2 (λ, φ) = , d2(λ, φ) = . k (2 sin φ)2λk! 0 (2 sin φ)2λ The ﬁrst Meixner-Pollaczek polynomials are

(λ) (λ) P0 (x; φ)(x) = 1,P1 (x; φ)(x) = 2λ cos φ + 2x sin φ, (λ) 2 2 2 2 P2 (x; φ)(x) = 2x sin φ + (1 + 2λ)x sin(2φ) + 2λ cos φ + λ cos(2φ), 2 2 2 2λ cos φ + λ EMeixPol(λ, φ) = −λ cot φ, EMeixPol (λ, φ) = (6.1) 2 sin2 φ 20 J.-R. Pycke

From (1.5), (2.2) and (6) we infer for the Meixner-Pollaczek p.d.f. the expression

(2 sin φ)2λ f˜(λ)(x, φ) = f (x)(x ∈ ) Γ(2λ) λ,φ R (2 sin φ)2λ = e(2φ−π)x|Γ(λ + ix)|2. 2πΓ(2λ)

From (2.10) we infer for the tail of the distribution and its m.g.f.

(2 sin φ)2λ f˜(λ)(x; φ) ∼ e2(φ−π)xx2λ−1 as x → ∞, Γ(2λ) (2 sin φ)2λ f˜(λ)(x, , φ) ∼ e2φx|x|2λ−1 as x → −∞, Γ(2λ) tMeixPol(λ,π) Ee < ∞ ⇐⇒ −2φ < t < 2(π − φ). (6.2)

Proposition 6.1 (From Meixner-Pollaczek to Laguerre). The following convergence 1 ( α+1 ) x (α) lim f˜ 2 (− ; φ) = f˜ (x) (6.3) φ→0+ 2φ 2φ holds, where f˜(α) is given by (8.3). Consequently (1.19) is valid.

Proof. Computations write, using (2.10),

α+1 ( α+1 ) x (2 sin φ) x α + 1 ix 2 f˜ 2 (− ; φ) = exp{(2φ − π)(− )}|Γ( − )| 2φ 2πΓ(α + 1) 2φ 2 2φ (2φ)α+1 πx x πx ∼ exp{−x + )}2π( )α exp{− )} 2πΓ(α + 1) 2φ 2φ 2φ

= 2φf˜α(x) as φ → 0+. So the result, as well as its corollary (1.17) is proved.

Proposition 6.2 (From Meixner-Pollaczek to Hermite). Assume that π (λ, φ) → (∞, ) and lim λ1/2 cos φ = 0. 2 Then √ (λ) x λ − λ cos φ lim f ( ; φ) = f˜0(x) (6.4) λ→∞ sin φ where f˜0 is given by (9.3). Consequently relation (1.18) holds. Probabilistic Askey scheme 21

Proof. On the first hand √ √ 2λ √ x λ − λ cos φ (2 sin φ) (2φ−π)( x λ−λ cos φ ) x λ − λ cos φ f (λ)( ; φ) = e sin φ |Γ(λ+i )|2 sin φ 2πΓ(2λ) sin φ √ Let us first apply formula (2.13) to z = λ + i(x λ − λ cos φ)/ sin φ. On the first hand elementary computations enable us to write

e−2Rz = e−2λ, √ x λ − λ cos φ |z|2R(z)−1 = (λ2 + ( )2)λ−1/2 sin φ √ λ2 2xλ λ cos φ x2λ = ( − + )λ−1/2 sin2 φ sin φ sin2 φ λ x sin(2φ) x2 = ( )2λ−1(1 − √ + )λ−1/2 sin φ λ λ λ x sin(2φ) x2 ∼ ( )2λ−1(1 − √ + )λ sin φ λ λ λ √ x2 ∼ ( )2λ−1 exp{−x λ sin(2φ) + x2 − sin2(2φ)} sin φ 2

λ 2 ∼ ( )2λ−1ex . sin φ On the other hand the asymptotic relation cos φ π arctan( + h) = − φ + h sin2 φ − h2 cos φ sin3 φ + o(h2) sin φ 2 which is valid as h → 0 enables us to write cos φ x − arg z = arctan( − √ ) sin φ sin φ λ π x sin φ x2 cos φ sin φ = − φ − √ − + o(λ−1) 2 λ λ √ x λ − λ cos φ π x sin φ x2 cos φ sin φ −2I(z) arg z = 2( )( − φ − √ − + o(1) sin φ 2 λ λ √ x λ − λ cos φ √ = ( )(π − 2φ) − 2x2 + 2x2 cos2 φ + 2x λ cos(φ) + o(1) sin φ √ x λ − λ cos φ = ( )(π − 2φ) − 2x2 + o(1). sin φ 22 J.-R. Pycke

Thus

√ √ x λ − λ cos φ λ −x2+( x λ−λ cos φ )(π−2φ) |Γ(λ + i )|2 ∼ 2πe−2λ( )2λ−1e sin φ . sin φ sin φ

Furthermore 1 ∼ (2π)−1/2e2λ(2λ)−2λ+1/2 Γ(2λ)

Consequently after simpliﬁcation we obtain

√ x λ − λ cos φ sin φ e−x2 f˜(λ)( ; φ) ∼ √ √ sin φ λ π which implies the desired results.

7 Jacobi polynomials and distributions

For Jacobi polynomials see [2] §6.3. They are are deﬁned for each k ∈ N by

! (α + 1)k −k, α + β + k + 1 1 − x Pk(α, β)(x) = 2F1 k! α + 1 2 and satisfy the orthogonality relations

Z 1 (α,β) (α,β) 2 Pk (x)P` (x)fα,β(x)dx = dk(α, β)δk` −1 with 2α+β+1Γ(α + k + 1)Γ(β + k + 1) d2 (α, β) = k k!(2k + α + β + 1)Γ(k + α + β + k + 1) whence 2α+β+1Γ(α + 1)Γ(β + 1) d2(α, β) = . (7.1) 0 Γ(α + β + 2) Probabilistic Askey scheme 23

The ﬁrst Jacobi polynomials and moments of Jac(α, β) are

(α,β) P0 (x) = 1, α + β + 2 α − β P (α,β)(x) = x + , 1 2 2 x2 P (α,β)(x) = [(1 + α)(2 + α) + (1 + β)(2 + β) + 2(2 + α)(2 + β)] 2 8 x + [(1 + α)(2 + α) + (1 + β)(2 + β)] 4 (1 + α)(2 + α) + (1 + β)(2 + β) − 2(2 + α)(2 + β) + , 8 β − α Jac(α, β) = . E α + β + 2 (7.2)

In view of (1.6), (2.2 and (7.1) the p.d.f associated with Jacobi polynomials is the Beta distribution on (−1, 1) deﬁned by

Γ(α + β + 2) f˜ (x) = f (x) α,β 2α+β+1Γ(α + 1)Γ(β + 1) α,β Γ(α + β + 2) = (1 − x)α(1 + x)β (−1 < x < 1) (7.3) 2α+β+1Γ(α + 1)Γ(β + 1) It is readily checked that

tJac(α,β) Ee < ∞ (t ∈ R). (7.4)

Proposition 7.1 (From Jacobi to Laguerre). Assume α > 0. Then the convergence

2 2x (α) lim f˜α,β(1 − ) = f˜ (x) (7.5) β→∞ β β hold and implies (1.19).

Proof. On the ﬁrst hand we obtain successively 2 2x Γ(α + β + 2) 2x 2x f˜ (1 − ) = ( )α(2 − )β β α,β β 2α+ββΓ(α + 1)Γ(β + 1) β β Γ(β + 2 + α) x = xα(1 − )β ∼ xαe−x β1+αΓ(β + 1) β which implies the desired convergences. 24 J.-R. Pycke

Proposition 7.2 (From Jacobi to Hermite). For each x ∈ R one has

1 (α,α) x lim √ f˜ (√ ) = f˜0(x) (7.6) α→∞ α α where f˜0 is deﬁned by (9.3). Consequently (1.20) holds. Proof. On using (2.7) simple computations give 1 x Γ(2α + 2) x2 √ f˜(α,α)(√ ) = (1 − )α α α α1/222α+1Γ(α + 1)2 α (π)−1/222α+1Γ(α + 1)Γ(α + 3 ) x2 = 2 (1 − )α α1/222α+1Γ(α + 1)2 α e−x2 ∼ √ π which implies the claimed convergence.

8 Laguerre polynomials and related distributions

(α) The well-known Laguerre polynomials are denoted by Lk and deﬁned in terms of hypergeometric function by ! (α) (α + 1)n −n L (x) = 1F1 x . k n! α + 1 They satisfy the orthogonality relations Z ∞ (α) (α) 2 Lk (x)L` (x)fα(x)dx = dk(α)δk` 0 with Γ(α + k + 1) d2 (α) = k k! whence 2 d0(α) = Γ(α + 1). (8.1) The ﬁrst Laguerre polynomials and moment of Lag(α) are

(α) (α) L0 (x) = 1,L1 (x) = −x + α + 1 x2 (α + 2)(α + 1) L(α)(x) = − (α + 2)x + 2 2 2 2 ELag(α) = α + 1, ELag (α) = (α + 1)(α + 2), VLag(α) = α + 1 (8.2) Probabilistic Askey scheme 25

In view of (1.7), (2.2) and (8.1), the p.d.f associated with Laguerre polynomials is

f (x) xαe−x f˜ (x) = α = (x > 0). (8.3) α Γ(α + 1) Γ(α + 1)

It is readily checked that

tJac(α) Ee < ∞ ⇐⇒ t < 1. (8.4)

Proposition 8.1 (From Laguerre to Hermite). For each x ∈ R both convergences √ √ lim 2αf˜α(x 2α + α) = f˜0(x) (8.5) α→∞ and (1.21) hold.

Proof. As α → ∞, √ √ √ √ √ (x 2α + α)αe−x 2α−α 2αf˜ (x 2α + α) = 2α α Γ(α + 1) √ √ √ √ √ ααex 2α−x2 e−x 2α−α √ ααex 2α−x2 e−x 2α−α e−x2 ∼ 2α √ ∼ 2α √ = √ e−α−1(α + 1)1/2+α 2π e−α−1eα1/2+α 2π π and the claimed result is proved.

9 Hermite polynomials and the normal distribution

For each k ∈ N the Hermite polynomials are deﬁned by ! k −k/2, −(k − 1)/2 2 Hk(x) = (2x) 2F0 − 1/x (9.1) − satisfy the orthogonality relation Z −x2 k √ e Hk(x)H`(x)dx = δk`2 k! π (9.2) R

2 k √ 2 √ so that dk = 2 k! π and d0 = π so that the corresponding p.d.f. is given by

f (x) e−x2 f˜ (x) = √0 = √ (9.3) 0 π π 26 J.-R. Pycke which is the normal distribution N(0, 1/2) as expected. The ﬁrst three polynomials and related moments are a

2 H0(x) = 1,H1(x) = 2x, H2(x) = 4x − 2, 1 Her = 0, Her2 = , Her = 1/2. (9.4) E E 2 V It is readlily checked that

tHer Ee < ∞ (t ∈ R). (9.5)

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Received. Probabilistic Askey scheme 27

Author information Jean-Renaud Pycke, Department of Mathematics, University of Evry, LPMA, 91037 Evry cedex (France), France. E-mail: [email protected]