Classical discrete d -orthogonal polynomials Naoures AYADI - - PowerPoint PPT Presentation

AIMS-VOLKSWAGEN STIFTUNG WORKSHOP, DOUALA, 2018.

Classical discrete d-orthogonal polynomials

Naoures AYADI nawres.essths@gmail.com Under the direction of Hamza CHAGGARA

• Lab. Mathematics-Physics, Special Functions and

Applications ESSTHS, SOUSSE UNIVERSITY, TUNISIA

October 11, 2018

1

Plan

Naoures AYADI | Classical discrete d-orthogonal polynomials

1

Plan

1 Preliminaries and Notations.

Naoures AYADI | Classical discrete d-orthogonal polynomials

1

Plan

1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs

Naoures AYADI | Classical discrete d-orthogonal polynomials

1

Plan

1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d-Orthogonality and Classical discrete d-OPSs

Naoures AYADI | Classical discrete d-orthogonal polynomials

1

Plan

1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d-Orthogonality and Classical discrete d-OPSs 4 Examples

Naoures AYADI | Classical discrete d-orthogonal polynomials

1

Plan

1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d-Orthogonality and Classical discrete d-OPSs 4 Examples

• ∆ω-Appell polynomial set: d-OPS of Charlier type

Naoures AYADI | Classical discrete d-orthogonal polynomials

1

Plan

1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d-Orthogonality and Classical discrete d-OPSs 4 Examples

• ∆ω-Appell polynomial set: d-OPS of Charlier type
• 2-OPS of Meixner type

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

• A PS, {Pn}n0, is monic if Pn = xn + an−1xn−1 + · · · + a0, n 0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

• A PS, {Pn}n0, is monic if Pn = xn + an−1xn−1 + · · · + a0, n 0.
• τb: The translation operator

(τbf)(x) = f(x − b), f ∈ P, b ∈ R.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

• A PS, {Pn}n0, is monic if Pn = xn + an−1xn−1 + · · · + a0, n 0.
• τb: The translation operator

(τbf)(x) = f(x − b), f ∈ P, b ∈ R.

• ∆ω, ∇ω: Hahn’s operators

∆ω(f)(x) = f(x + ω) − f(x) ω , and ∇ωf(x) = f(x) − f(x − ω) ω = ∆−ωf(x); ω = 0, f ∈ P.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

• A PS, {Pn}n0, is monic if Pn = xn + an−1xn−1 + · · · + a0, n 0.
• τb: The translation operator

(τbf)(x) = f(x − b), f ∈ P, b ∈ R.

• ∆ω, ∇ω: Hahn’s operators

∆ω(f)(x) = f(x + ω) − f(x) ω , and ∇ωf(x) = f(x) − f(x − ω) ω = ∆−ωf(x); ω = 0, f ∈ P.

• ∆1 ≡ ∆ and ∇1 ≡ ∇.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

• A PS, {Pn}n0, is monic if Pn = xn + an−1xn−1 + · · · + a0, n 0.
• τb: The translation operator

(τbf)(x) = f(x − b), f ∈ P, b ∈ R.

• ∆ω, ∇ω: Hahn’s operators

∆ω(f)(x) = f(x + ω) − f(x) ω , and ∇ωf(x) = f(x) − f(x − ω) ω = ∆−ωf(x); ω = 0, f ∈ P.

• ∆1 ≡ ∆ and ∇1 ≡ ∇.
• L, ∆ωf = −∇ωL, f, L ∈ P′, f ∈ P.

Naoures AYADI | Classical discrete d-orthogonal polynomials

2

Preliminaries and Notations

• P: The vector space of polynomials with coefficients in C and P′ its

dual.

• PS: The polynomial set {Pn}n0 ∈ P, such that deg Pn = n for all n.
• L, f: The duality brackets between P′ and P

(L ∈ P′ and f ∈ P).

• A PS, {Pn}n0, is monic if Pn = xn + an−1xn−1 + · · · + a0, n 0.
• τb: The translation operator

(τbf)(x) = f(x − b), f ∈ P, b ∈ R.

• ∆ω, ∇ω: Hahn’s operators

∆ω(f)(x) = f(x + ω) − f(x) ω , and ∇ωf(x) = f(x) − f(x − ω) ω = ∆−ωf(x); ω = 0, f ∈ P.

• ∆1 ≡ ∆ and ∇1 ≡ ∇.
• L, ∆ωf = −∇ωL, f, L ∈ P′, f ∈ P.
• (µn)n := u, xn, n 0: The moment sequence of u ∈ P′.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Preliminaries and Notations

Definition

Let {Pn}n0 be a monic PS. We call dual sequence of {Pn}n0, the sequence of linear forms {Fn}n0 defined by Fn, Pm = δn,m; n, m 0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

3

Preliminaries and Notations

Definition

Let {Pn}n0 be a monic PS. We call dual sequence of {Pn}n0, the sequence of linear forms {Fn}n0 defined by Fn, Pm = δn,m; n, m 0.

Lemma

Let L ∈ P′, k 1, and {Pn}n0 ∈ P. In order that L satisfies L, Pk−1 = 0, and L, Pn = 0; n k, (1) it is necessary and sufficient that there exists λν ∈ C, 0 ν k − 1, λk−1 = 0, such that L =

k−1

• ν=0

λν Fν. (2)

Naoures AYADI | Classical discrete d-orthogonal polynomials

4

Orthogonality

Definition

A PS, {Pn}n0, is said to be an orthogonal polynomial sequence (OPS) with respect to a functional L if: L, PnPm = Knδn,m (Kn = 0), (3) for n, m = 0, 1, 2, · · · .

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Orthogonality

Definition

A PS, {Pn}n0, is said to be an orthogonal polynomial sequence (OPS) with respect to a functional L if: L, PnPm = Knδn,m (Kn = 0), (3) for n, m = 0, 1, 2, · · · .

The functional L is called regular.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Orthogonality

Definition

Orthogonal polynomials of discrete variables are polynomials Pn, n = 0, 1... that satisfy the orthogonality relation

• i

Pn(xi)Pm(xi)ρ(xi) = 0, n = m, (4) where ρ is a positive weight function. The summation is done on the values of x which satisfy a xi < b where xi+1 = xi + 1, a and b are finite or infinite.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Classical OPSs

Lemma

Let {Pn}n0 be a monic PS and {Fn}n0 its dual sequence. We consider {Qω

n }n0 defined by

Qω

n (x) = ∆ωPn+1(x)

n + 1 , n 0, (5) and

• Fn
• n0 its associated dual sequence, then we have

∇ω Fn = −(n + 1)Fn+1, n 0. (6)

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Classical OPSs

Lemma

Let {Pn}n0 be a monic PS and {Fn}n0 its dual sequence. We consider {Qω

n }n0 defined by

Qω

n (x) = ∆ωPn+1(x)

n + 1 , n 0, (5) and

• Fn
• n0 its associated dual sequence, then we have

∇ω Fn = −(n + 1)Fn+1, n 0. (6)

Definition

A PS {Pn}n0 is said to be classical discrete orthogonal polynomial

• r ∆ω-Classical OPS if {Qω

n }n0 is also OPS.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Characterizations of ∆ω-classical OPS

Hahn (1949): {∆ωPn}n0 is also OPS

• Naoures AYADI | Classical discrete d-orthogonal polynomials

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Characterizations of ∆ω-classical OPS

Hahn (1949): {∆ωPn}n0 is also OPS

• Hildebrandt (1931): Pearson equation:

∇ω(σ(x)ρ(x)) = τ(x)ρ(x).

• Naoures AYADI | Classical discrete d-orthogonal polynomials

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Characterizations of ∆ω-classical OPS

Hahn (1949): {∆ωPn}n0 is also OPS

• Hildebrandt (1931): Pearson equation:

∇ω(σ(x)ρ(x)) = τ(x)ρ(x).

• Tricomi (1948): Rodrigues Formula

Pn(x) = Bn ρ(x)∇n

ω

• ρ(x)

n−1

• k=0

σ(x + ωk)

• .
• Naoures AYADI | Classical discrete d-orthogonal polynomials

7

Characterizations of ∆ω-classical OPS

Hahn (1949): {∆ωPn}n0 is also OPS

• Hildebrandt (1931): Pearson equation:

∇ω(σ(x)ρ(x)) = τ(x)ρ(x).

• Tricomi (1948): Rodrigues Formula

Pn(x) = Bn ρ(x)∇n

ω

• ρ(x)

n−1

• k=0

σ(x + ωk)

• .
• Bochner (1929): Sturm-Liouville linear second-order difference

equation σ(x)∆ω∇ωy(x) + τ(x)∆ωy(x) + λny(x) = 0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Classical OPSs

Theorem (F. Abdelkarim and P . Maroni 1997)

We have the following equivalence: (a) The monic PS, {Pn}n0, is ∆ω-classical OPS with respect to F. (b) The functional F satisfies the equation ∇ω(ΦF) + ΨF = 0, (7) with deg(Φ) 2 et deg(Ψ) = 1; Ψ′(0) − 1 2Φ′′(0)n = 0, n 1. (8)

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Classical OPSs

In the case ω = 1, we have:

• The Charlier polynomial {Cn(., a)}n0 satisfy the orthogonality

relation, for 0 m n:

+∞

• x=0

ax x !e−aCn(x; a)Cm(x; a) = a−nn !δn,m; a > 0. The functional associated with the monic Charlier OPS verify Pearson type equation: ∇ωF + a−1(x − a)F = 0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

9

∆ω-Classical OPSs

In the case ω = 1, we have:

• The Charlier polynomial {Cn(., a)}n0 satisfy the orthogonality

relation, for 0 m n:

+∞

• x=0

ax x !e−aCn(x; a)Cm(x; a) = a−nn !δn,m; a > 0. The functional associated with the monic Charlier OPS verify Pearson type equation: ∇ωF + a−1(x − a)F = 0.

• The Meixner polynomial {Mn(., β, c)}n0 satisfy the orthogonality

relation, for 0 m n:

+∞

• x=0

(β)x x ! cxMn(x; β, c)Mm(x; β, c) = c−nn ! (β)n(1 − c)β δn,m; β > 0, 0 < c < 1. The functional associated with the monic Meixner verify Pearson type equation: ∇ω((1 + β + x)F) − ((1 − c−1)x + β + 1)F = 0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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d-Orthogonality

Definition ( V. Iseghem and P . Maroni )

Let d be a positive integer and let {Pn}n0 be a PS in P. {Pn}n0 is called a d-orthogonal polynomial set (d-OPS for shorter) with respect to the d-dimensional functional vector U = t(u0, u1, · · · , ud−1) if it satisfies the following conditions

• uk, PmPn = 0, m > nd + k, n 0,

uk, PnPnd+k = 0, n 0, (9) for each k ∈ {0, 1, · · · , d − 1}.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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d-Orthogonality

Theorem (V. Iseghem and P . Maroni 1989)

For each monic PS {Pn}n0, the following statements are equivalent (a) The sequence {Pn}n0 satisfies a (d + 1)-order recurrence relation: Pn+d+1(x) = (x − βn+d)Pn+d(x) −

d−1

• ν=0

γd−1−ν

n+d−ν Pn+d−1−ν(x); n 0,

(10) with initial conditions      P0(x) = 1; P1(x) = x − β0, Pn(x) = (x − βn−1)Pn−1(x) −

n−2

• ν=0

γd−1−ν

n−1−ν Pn−2−ν(x), 2 n d,

(11) and the regularity conditions γ0

n+1 = 0;

n 0. (b) The sequence {Pn}n0 is d-orthogonal with respect to F = t(F0, F1, ..., Fd−1), formed by the d first terms of its dual sequence {Fn}n0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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d-Orthogonality

(c) For every (n, ν), n 0, 0 ν d − 1, there exists d polynomials Λµ(n, ν), 0 µ d − 1 such that Fnd+ν =

d−1

• µ=0

Λµ(n, ν)Fµ, n 0, 0 ν d − 1. (12) where deg Λν(n, ν) = n, 0 ν d − 1, deg Λµ(n, ν) n, 0 µ ν − 1 if 1 ν d − 1, deg Λµ(n, ν) n − 1, ν + 1 µ d − 1 if 0 ν d − 2.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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d-Orthogonality

Lemma

For each PS {Pn}n0 d-orthogonal with respect to F = t (F0, F1, ...., Fd−1), the following statements are equivalent (a) There exists L ∈ P′ and s ∈ N∗, such that:

• L, Pn(x) = 0;

n s, L, Ps−1(x) = 0. (b) There exists L ∈ P′, s 1 and d polynomials φα; 0 α d − 1, such that: L =

d−1

• α=0

φαFα. with the following properties: If s −1 = qd +r, 0 r d −1, we have      deg φr = q, if d 2, deg φα q, 0 α r − 1 if r 1, deg φα q − 1, r + 1 < α d − 1 if r = d − 1.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Classical d-OPSs

Definition

A d-OPS, {Pn}n0, is said to be ∆ω-classical if the PS {Qω

n }n0 is

also d-OPS.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Characterization of ∆ω-Classical d-OPSs

Theorem

The following assertions are equivalent (a) The monic sequence {Pn}n0 is ∆ω-classical d-OPS. (b) The functional F = t (F0, F1, ..., Fd−1) satisfies the following matrix equation ∇ω(ΦF) + ΨF = 0, (13) where Ψ and Φ are two d × d matrices of polynomials Ψ(x) =        1 · · · 2 · · · · · · · · · · · · · · · · · · · · · · · · d − 1 ψ(x) ξ1 ξ2 ξ3 · · · ξd−2 ξd−1        (14) with ψ(x) = d γ0

1

P1(x) ; and ξµ = −dγd−µ

1

γ0

1

, 1 µ d − 1,

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Characterization of ∆ω-Classical d-OPSs

and Φ(x) =        φ0

0(x)

φ1

0(x)

· · · · · · φd−1 (x) · · · · · · · · · · · · · · φ0

d−2

φ1

d−2(x)

· · · · · · φd−1

d−2(x)

φ0

d−1(x)

φ1

d−1

· · · · · · φd−1

d−1(x)

       , (15) with φν

α are polynomials such that

deg φν

α 1,

0 ν α + 1 if 0 α d − 2. deg φν

α = 0,

α + 2 ν d − 1 if 0 α d − 3. deg φ0

d−1 2 and deg φν d−1 1,

1 ν d − 1.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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Characterization of ∆ω-Classical d-OPSs

and Φ(x) =        φ0

0(x)

φ1

0(x)

· · · · · · φd−1 (x) · · · · · · · · · · · · · · φ0

d−2

φ1

d−2(x)

· · · · · · φd−1

d−2(x)

φ0

d−1(x)

φ1

d−1

· · · · · · φd−1

d−1(x)

       , (15) with φν

α are polynomials such that

deg φν

α 1,

0 ν α + 1 if 0 α d − 2. deg φν

α = 0,

α + 2 ν d − 1 if 0 α d − 3. deg φ0

d−1 2 and deg φν d−1 1,

1 ν d − 1. In addition, if we write

• ψ(x) = ψ′(0)x + ψ(0),

φ0

d−1(x) = 1 2(φ0 d−1)′′(0)x2 + (φ0 d−1)′′(0)x + φ0 d−1(0)

φα+1

α

(x) = (φα+1

α

)′(0)x + φα+1

α

(0), 0 α d − 2 . Then 1 2 (φ0

d−1)′′(0) = ψ′(0)

m + 1 , m 0, ψ′(0) = 0. (16) (φα+1

α

)′(0) = α + 1 m + 1 , m 0, for 0 α d − 2. (17)

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Classical d-OPSs

Corollary

Let {Pn}n0 be a ∆ω-classical d-OPS with respect to F and ∇ω(ΦF) + ΨF = 0. For each k = 1, 2, · · · , let Fk be the functional

• Fk =

 

k−1

• ν=0

τ−νωΦ   F, and the sequence of monic polynomials Qω

n,k(x) =

∆k

ωPn+k(x)

(n + 1)(n + 2) · · · (n + k) . Then for each k 1,

• Qω

n,k

• n0 is also d-OPS with respect to
• Fk. Moreover,

∇ω

• Φk

Fk + Ψk Fk = 0, with Φk and Ψk are two polynomials matrices defined by Φk(x) = (τ−kωΦ)(x), and Ψk(x) = Ψ(x) −

k−1

• ν=0

(∆ω ◦ τ−νωΦ) (x) Therefore, this proves that

• Qω

n,k

• n0 is also ∆ω-classical.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Appell polynomial set: d-OPS of Charlier type

• A PS {Pn}n0 is said to be ∆ω-Appell if and only if it verifies:

∆ωPn+1(x) = (n + 1)Pn(x), n 0. (18)

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Appell polynomial set: d-OPS of Charlier type

• A PS {Pn}n0 is said to be ∆ω-Appell if and only if it verifies:

∆ωPn+1(x) = (n + 1)Pn(x), n 0. (18)

• ∆ω-Appell PS −

→ ∆ω-classical PS, and Qω

n (x) = Pn(x); n 0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Appell polynomial set: d-OPS of Charlier type

• A PS {Pn}n0 is said to be ∆ω-Appell if and only if it verifies:

∆ωPn+1(x) = (n + 1)Pn(x), n 0. (18)

• ∆ω-Appell PS −

→ ∆ω-classical PS, and Qω

n (x) = Pn(x); n 0.

• The functional vector of the ∆ω-Appell PS satisfies the matrix

equation (13) where Φ and Ψ are given by

Ψ(x) =          1 · · · 2 · · · · · · · · · · · · · · · · · · · · · · d − 1

d(x−β0+ω) γ0

1

ξ1 ξ2 · · · ξd−2 ξd−1          , Φ(x) =      1 · · · 1 · · · · · · · · · · · · · · · · · · · · 1     

with ξµ =

−dγd−µ

1

γ0

1

, 1 µ d − 1.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Appell polynomial set: d-OPS of Charlier type

• Ben Cheikh and Zaghouani (2003) proved that the only d-OPSs

∆ω-Appell PSs, are generated by: exp(Hd(t))(1 + ωt)

x ω =

+∞

• n=0

Pn(x) tn n !, Hd is a polynomial of degree d. The case, Hd(t) = −atd and ω = 1, coincides with the d-OPS of Charlier type {Cn(.; a, d)}n0.

Naoures AYADI | Classical discrete d-orthogonal polynomials

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∆ω-Appell polynomial set: d-OPS of Charlier type

• Ben Cheikh and Zaghouani (2003) proved that the only d-OPSs

∆ω-Appell PSs, are generated by: exp(Hd(t))(1 + ωt)

x ω =

+∞

• n=0

Pn(x) tn n !, Hd is a polynomial of degree d. The case, Hd(t) = −atd and ω = 1, coincides with the d-OPS of Charlier type {Cn(.; a, d)}n0.

• The functional vector associated to the ∆-classical d-OPS of

Charlier type verifies the matrix equation (13), with Ψ is Ψ(x) =         1 · · · 2 · · · · · · · · · · · · · · · · · · · · · · · · d − 1

x ad!

· · · −1         .

Naoures AYADI | Classical discrete d-orthogonal polynomials

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2-OPS of Meixner type

• The 2-OPS of Meixner type is given by

Mβ1,β2

n

(x; c) = 2F2 −n, −x β1, β2 ; 1 c

• where pFq is the hypergeometric series defined by

pFq

• a1, a2, ..., ap

b1, b2, ..., bq ; z

• :=

∞

• m=0

(a1)m....(ap)m (b1)m.....(bq)m zm m !, the Pochhammer’s symbol (a)m, defined by: (a)m =

• 1

if m = 0, a(a + 1).....(a + m − 1) if m = 1, 2, 3....

Naoures AYADI | Classical discrete d-orthogonal polynomials

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2-OPS of Meixner type

• The 2-OPS of Meixner type is given by

Mβ1,β2

n

(x; c) = 2F2 −n, −x β1, β2 ; 1 c

• where pFq is the hypergeometric series defined by

pFq

• a1, a2, ..., ap

b1, b2, ..., bq ; z

• :=

∞

• m=0

(a1)m....(ap)m (b1)m.....(bq)m zm m !, the Pochhammer’s symbol (a)m, defined by: (a)m =

• 1

if m = 0, a(a + 1).....(a + m − 1) if m = 1, 2, 3....

• The monic polynomial of Meixner type is given by
• Mβ1,β2

n

(x; c) = (β1)n(β2)ncnMβ1,β2

n

(x; c).

Naoures AYADI | Classical discrete d-orthogonal polynomials

21

2-OPS of Meixner type

• Via the Maple software, and specially, the sumrecursion command
• f the hsum17 package, we obtain the third-order recurrence relation
• f
• Mβ1,β2

n

(.; c)

• n0:
• Mβ1,β2

n+3 (x; c) = (x−αn+2)

Mβ1,β2

n+2 (x; c)−γn+2

Mβ1,β2

n+1 (x; c)−δn+1

Mβ1,β2

n

(x; c), (19) with      αn+2 = −c(2 + n) [2(β1 + β2) + β1β2 + (3n + 4)] γn+2 = c(2 + n)(β1 + 1 + n)(β2 + 1 + n)(−1 + c(4 + 3n) + c(β1 + β2)) δn+1 = −c3(n + 2)(n + 1)(β1 + 1 + n)(β2 + 1 + n)(β1 + n)(β2 + n) and

• Mβ1,β2

1

(x; c) = x − α0; Mβ1,β2

2

(x; c) = (x − α1)(x − α0) − γ1. (20)

Naoures AYADI | Classical discrete d-orthogonal polynomials

22

2-OPS of Meixner type

• The sequence
• Mβ1,β2

n

(.; c)

• n0 is ∆-classical 2-OPS, since

Qn(x) = ∆ Mβ1,β2

n+1 (x; c)

n + 1 = Mβ1+1,β2+1

n

(x; c). (21)

Naoures AYADI | Classical discrete d-orthogonal polynomials

22

2-OPS of Meixner type

• The sequence
• Mβ1,β2

n

(.; c)

• n0 is ∆-classical 2-OPS, since

Qn(x) = ∆ Mβ1,β2

n+1 (x; c)

n + 1 = Mβ1+1,β2+1

n

(x; c). (21)

• Let F = t(F0, F1) be the functional vector associated with the PS,
• Mβ1,β2

n

(.; c)

• n0. Then, F satisfies the following matrix equation

∇(ΦF) + ΨF = 0, with Φ(x) = φ0

0(x)

φ1

0(x)

φ0

1(x)

φ1

1(x)

• and

Ψ(x) = 1 ψ(x) ξ1

• Naoures AYADI | Classical discrete d-orthogonal polynomials

23

2-OPS of Meixner type

where ψ(x) = 2 δ1

• M1(x; β, c) = −

1 c3(β1 + 1)(β2 + 1)(x + c)

ξ1 = −2γ1 δ1 = 1 c(β1 + 1)(β2 + 1) −1 c + 2 (β1 + 1)(β2 + 1) + β1β2 − (β1 + 1)(β2 + 1)

• and

φ0

0(x) = −

5 cβ1β2 x + 6, φ1

0(x) = 4(cβ1β2 − 1) +

10c (β1 + 1)(β2 + 1) − 5c(β1 + 1)(β2 + 1), φ0

1(x) = [(2c(β1 + β2) − 2cβ1β2 − 2)δ−1 1

− γ2 δ2δ1 ]x + γ2α0 − δ1 δ2δ1 , φ1

1(x) = 1 + 2c(β1 + β2) − 2cβ1β2 − 2 − δ−1 1 γ1 + γ2γ1 − α1δ1

δ2δ1 + δ−1

2 x.

Naoures AYADI | Classical discrete d-orthogonal polynomials

24

References

[1] T.S. CHIHARA, An introduction to orthogonal polynomials; Gordon and Breach, New-York, London, Paris, 1978. [2] F . ABDELKARIM and P . MARONI, The Dω-classical orthogonal polynomials; Result. Math. 32 (1997), 1-28. [3] P . MARONI, L ’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux; Ann. Fac. Sci. Toulouse Math, 10 (1989), 105-139. [4]

• K. DOUAK and P

. MARONI, Une caractérisation des polynômes d-orthogonaux "Classiques" de dimension d; J. Approx. Theory, 82 (1995), 177-204. [5]

• A. Boukhemis, On the classical 2-orthogonal polynomials

sequences of Sheffer-Meixner type; Cubo. A Mathematical

• Journal. 7 (2005), 39-55.

Naoures AYADI | Classical discrete d-orthogonal polynomials

Thank you for your attention