Structure relations of classical multiple orthogonal polynomials by a generating function

J. Korean Math. Soc. 50 (2013), No. 5, pp. 1067–1082 http://dx.doi.org/10.4134/JKMS.2013.50.5.1067

STRUCTURE RELATIONS OF CLASSICAL MULTIPLE ORTHOGONAL POLYNOMIALS BY

A GENERATING FUNCTION

Dong Won Lee

Abstract. In this paper, we will find some recurrence relations of classi- cal multiple OPS between the same family with different parameters us- ing the generating functions, which are useful to find structure relations and their connection coefficients. In particular, the differential-difference equations of Jacobi-Pi˜ neiro polynomials and multiple Bessel polynomials are given.

1. Introduction

Let r ≥ 2 be a fixed positive integer, ~n = (n 1 , n 2 , . . . , n r ) ∈ N ^r ₀ a multi- index, and e i = (0, . . . , 1, . . . , 0) the i-th standard unit vector in R ^r with e = P r

i=1 e i . For any vector ~ α = (α 1 , α 2 , . . . , α r ) and t = (t 1 , t 2 , . . . , t r ), we let

|t| = t 1 + t 2 + · · · + t r and the product α · t = α 1 t 1 + α 2 t 2 + · · · + α r t r . A sequence {P _{~ n} (x)} ^∞ _{|~ n|=0} of polynomials is called a multiple orthogonal poly- nomial system (multiple OPS) if

(i) deg(P ~ n ) = |~n|;

(ii) there exist r positive weights w i such that for i = 1, 2, . . . , r, Z ∞

−∞

x ^k P ~ n (x)w i (x)dx = 0 for k = 0, 1, 2, . . . , n i − 1.

The multiple OPS was originated from the paper of Angelesco in dealing with simultaneous Pad´e approximants ([1]). These families of polynomials attracted big interest in the area of simultaneous Pad´e approximation, random matrix, asymptotics, number theory, and so on (see [6, 7, 9, 10, 11] for recent relevant references and therein).

Recently many results are obtained on so-called classical multiple OPS’s whose orthogonalizing weights w i are classical. More precisely, they are multi- ple Hermite polynomials, multiple Laguerre I polynomials, multiple Laguerre

Received November 30, 2012.

2010 Mathematics Subject Classification. 33C45, 42C05.

Key words and phrases. multiple orthogonal polynomial, classical multiple orthogonal polynomial, recurrence relation, generating function.

c

2013 The Korean Mathematical Society

1067

II polynomials, Jacobi-Pi˜ neiro polynomials, and multiple Bessel polynomials (see [2, 3, 16, 21] and references therein).

Among the classical multiple OPS’s the generating functions are developed for three families of them, which are the case of multiple Hermite polynomials, multiple Laguerre I polynomials, and multiple Laguerre II polynomials ([13]).

For these multiple OPS’s the orthogonalizing weights and the generating func- tions are as follow.

(a) multiple Hermite polynomials {H _{~ n} ^{(~ α)} (x)} ^∞ _{|~ n|=0} : The orthogonalizing we- ights are w i (x) = e ^{δ 2 x 2 +α i x} on (−∞, ∞), where δ < 0 and α i 6= α j for i 6= j. The generating function is

G ^(α) (x, t) = e ^{δx|t|+ δ 2 |t| 2 +|~ α·t|} , that means for x ∈ R and t i ∈ R for i = 1, 2, . . . , r,

G ^(α) (x, t) =

∞

X

~ n=0

H _{~ n} ^{(~ α)} (x) t ^{~ n}

~n! .

(b) multiple Laguerre I polynomials {L ^(~ _{~ n} ^α;β) (x)} ^∞ _{|~ n|=0} : The orthogonalizing weights are w i (x) = x ^{α i} e ^βx on (0, ∞), where α i > −1, β < 0, and α i − α j ∈ Z for i 6= j. The generating function is /

G ^{(~ α;β)} (x, t) =

r

Y

i=1

1

(1 − t i ) ^{α i +1} e ^βx



Qr i=1(1−ti) 1 −1

 , that means for x ∈ (0, ∞) and |t i | < 1 for i = 1, 2, . . . , r,

G ^{(~ α;β)} (x, t) =

∞

X

~ n=0

L _~ ^(~ _n ^α;β) (x) t ^{~ n}

~n! .

(c) multiple Laguerre II polynomials {L _~ ^(α;~ _n ^β) (x)} ^∞ _{|~ n|=0} : The orthogonalizing weights are w i (x) = x ^α e ^{β i x} on (0, ∞), where α > −1, β i < 0, and β i 6= β j for i 6= j. The generating function is

G ^{(α;~ β)} (x, t) = 1

(1 − |t|) ^α+1 e ^{|~ 1−|t| β·t|x} , that means for x ∈ (0, ∞) and |t| < 1,

G ^{(α;~ β)} (x, t) =

∞

X

~ n=0

L _~ ^(α;~ _n ^β) (x) t ^{~ n}

~n! .

Here, we used multi-index notations ~n! = n 1 !n 2 ! · · · n r !, t ^{~ n} = t ⁿ ₁ ¹ t ⁿ ₂ ² · · · t ⁿ _r ^r , and P ∞

~ n=0 = P ∞ n 1 =0

P ∞

n 2 =0 · · · P ∞ n r =0 .

By the generating function, many properties of multiple Hermite polynomi-

als and the multiple Laguerre polynomials were developed such as differential-

difference relation and differential equations (see [13]).

On the other hand, recurrence relations of Jacobi-Pi˜ neiro polynomials and multiple Bessel polynomials are not rigorously studied until now. In order to get their recurrence relations, we need their generating functions. Recently we found the generating function for them in the case of r = 2 by Lagrange expansion method (see [14]). More precisely, the author proved:

(d) Jacobi-Pi˜ neiro polynomials {P n ^(α 1 ,n ^{1 ,α} 2 ^{2 ;α)} (x)} ^∞ _{n 1 +n 2 =0} : The orthogonal- izing weights are w i (x) = x ^{α i} (x − 1) ^α (i = 1, 2) on (0, 1), where α 1 , α 2 , α > −1 and α 1 − α 2 ∈ Z. The generating function is /

G ^{(α 1 ,α 2 ;α)} (x, t) = ¹

[(1 + t 1 − 2t 1 z)(1 + t 2 − 2t 2 z) − t 1 t 2 z ² ]

(1 + t 1 − t 1 z) ^−α

¹

(1 + t 2 − t 2 z) ^−α

²

[(1 − t 1 z)(1 − t 2 z) − t 1 t 2 z] ^α , that means for |t 1 | < 1, |t 2 | < 1, and x ∈ (0, 1),

G ^{(α 1 ,α 2 ;α)} (x, t) =

∞

X

n 1 =0

∞

X

n 2 =0

P _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! ,

where z is a solution of z(1 + t 1 − t 1 z)(1 + t 2 − t 2 z) = x with z → x as t 1 , t 2 → 0.

(e) multiple Bessel polynomials {B ^(α n 1 ¹ ,n ^,α 2 ^{2 ;γ)} (x)} ^∞ _{n 1 +n 2 =0} : The orthogonal- izing weights are w i (x) = x ^{α i} e ^{γ x} (i = 1, 2) on the unit circle in complex plane, where α 1 , α 2 > −1, γ 6= 0, and α 1 − α 2 ∈ Z. The generating / function is

G ^{(α 1 ,α 2 ;γ)} (x, t) = (1 − t 1 z) ^{−α 1} (1 − t 2 z) ^{−α 2}

(1 − 2t 1 z)(1 − 2t 2 z) − t 1 t 2 z ² e ^{γ ( 1 z − x 1 )} , that means for |t 1 | < 1, |t 2 | < 1, and |x| = 1 on complex plane,

G ^{(α 1 ,α 2 ;γ)} (x, t) =

∞

X

n 1 =0

∞

X

n 2 =0

B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 ;γ)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! ,

where z is a solution of z(1 − t 1 z)(1 − t 2 z) = x with z → x as t 1 , t 2 → 0.

There are tremendous recurrence relations for orthogonal polynomials such as three term recurrence relation, differential-difference equation, and so on.

We refer to [8, 17, 18]. The recurrence relation plays a key role in applications of orthogonal polynomials in area of rational approximation, quadrature formula, special functions, combinatorics, differential equations and so on.

For multiple OPS’s, many recurrence relations are obtained as an extension of orthogonal polynomials and so they would be used in many areas of multiple OPS’s (see [4, 5, 12, 20] and references therein). Most of these papers treated the recurrence relations with the same polynomials or differential-difference relation of the same polynomials. Comparing to ordinary classical OPS, we can deduce that the recurrence relations of the same family of classical multiple OPS with different parameter will also play an important role in investigating the properties of multiple OPSs.

In this paper, we find some recurrence relations of classical multiple OPS be-

tween the same family with different parameters using the generating function,

which are useful to find the structure relations and their connection coefficients.

In particular, the differential-difference equations of Jacobi-Pi˜ neiro polynomials and multiple Bessel polynomials are given.

2. Multiple Hermite polynomials

Using the identities of G ^{(~ α)} (x, t) for multiple Hermite polynomials

∂

∂x G ^{(~ α)} (x, t) = δ|t|G ^{(~ α)} (x, t)

and ∂

∂t i

G ^{(~ α)} (x, t) = (δx + α i )G ^{(~ α)} (x, t) + ∂

∂x G ^{(~ α)} (x, t), the author found ([13, Theorem 2.4]) differential-difference equations

d

dx H _{~ n} ^{(~ α)} (x) = δ

r

X

j=1

H _{~ n−e} ^{(~ α)} _j (x) = H _{~ n+e} ^{(~ α)} _i (x) − (δx + α i )H _{~ n} ^(α) (x), i = 1, 2, . . . , r.

These relations can be regarded as generalizations of the relation H _n ^′ (x) = nH n−1 (x) = 2xH n (x) − 2H n+1 (x),

where {H n (x)} ^∞ _n=0 is the monic Hermite polynomials orthogonal with respect to w(x) = e ^{−x 2} on (−∞, ∞).

By a simple calculation we have an identity (2.1) G ^{(~ α+e i )} (x, t) = e ^{t i} G ^{(~ α)} (x, t),

from which a new recurrence relation for multiple Hermite polynomials imme- diately follows.

Theorem 2.1. Let {H _{~ n} ^{(~ α)} (x)} ^∞ _{|~ n|=0} be the multiple Hermite polynomials. Then we have for i = 1, 2, . . . , r,

H _{~ n} ^{(~ α+e i )} (x) =

n i

X

j=0

n i

j



H _{~ n−je} ^(α) _i (x).

Proof. From the definition of generating function, we have for i = 1, 2, . . . , r, (2.2) G ^{(~ α+e i )} (x, t) =

∞

X

~ n=0

H _{~ n} ^{(~ α+e i )} (x) t ^{~ n}

~n! . On the other hand,

e ^{t i} G ^{(~ α)} (x, t) =





∞

X

j=0

t ^j _i j!





∞

X

~ n=0

H _{~ n} ^{(~ α)} (x) t ^{~ n}

~n!

!

= X _′





∞

X

n i =0

∞

X

j=0

H _{~ n} ^{(~ α)} (x) t ⁿ _i ^{i +j} n i !j!





t ⁿ ₁ ¹ · · · t ⁿ _i−1 ⁱ⁻¹ t ⁿ _i+1 ⁱ⁺¹ · · · t ⁿ _r ^r

n 1 ! · · · n i−1 !n i+1 ! · · · n r ! ,

where P _′

:= P ∞ n 1 =0

P ∞

n 2 =0 · · · P ∞ n i−1 =0

P ∞

n i+1 =0 · · · P ∞

n r =0 . By change of variables we have

∞

X

n i =0

∞

X

j=0

H _{~ n} ^{(~ α)} (x) t ⁿ _i ^{i +j} n i !j! =

∞

X

n i =0 n i

X

j=0

H _{~ n−je} ^{(~ α)} _i (x) t ⁿ _i ⁱ (n i − j)!j!

=

∞

X

n i =0 n i

X

j=0

n i

j



H _{~ n−je} ^{(~ α)} _i (x) t ⁿ _i ⁱ n i ! , so that

(2.3) e ^{t i} G ^{(~ α)} (x, t) =

∞

X

~ n=0 n i

X

j=0

n i

j



H _{~ n−je} ^{(~ α)} _i (x) t ^{~ n}

~n! .

From the equation (2.1), the conclusion follows by comparing the coefficients

of (2.2) and (2.3). 

In particular, if r = 2, then H _n ^(α ₁ ¹ _,n ^+1,α ₂ ^{2 )} (x) =

n 1

X

j=0

n 1

j



H _n ^(α _{1 −j,n} ^{1 ,α 2 )} ₂ (x)

and

H _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 +1)} (x) =

n 2

X

j=0

n 2

j



H _n ^(α ₁ ¹ _,n ^,α _{2 −j} ^{2 )} (x).

Applying Theorem 2.1 iteratively we obtain H _{~ n} ^{(~ α+e)} (x) =

~ n

X

~ k=0

~n

~k

 H _{~ n−} ^{(~ α) P} r

j=1 k j e j (x), where ~k = (k 1 , k 2 , . . . , k r ). Here, we used the notations P ~ n

~ k=0 = P n 1

k 1 =0

P n 2

k 2 =0

· · · P n r

k r =0 and ^{~ n} ~ k
= ⁿ _{k 1} ¹
_{n 2}

k 2
· · · ⁿ _{k r} ^r
.

The recurrence relation in Theorem 2.1 is quite interesting because we could not find a similar relation for Hermite polynomials. Hence, the relation is a new property that distinguishes multiple Hermite polynomials from Hermite polynomials.

3. Multiple Laguerre I polynomials

Using the identities of G ^{(~ α;β)} (x, t) for multiple Laguerre I polynomials

∂

∂x G ^{(~ α;β)} (x, t) = β 1 Q r

j=1 (1 − t j ) − 1

!

G ^{(~ α;β)} (x, t) and

(3.1) ∂

∂t i

G ^{(~ α;β)} (x, t) = βx + α i + 1 1 − t i

G ^{(~ α;β)} (x, t) + x 1 − t i

∂

∂x G ^{(~ α;β)} (x, t),

the author found ([13, Theorem 2.7]) differential-difference equations d

dx L ^(~ _{~ n} ^α;β) (x) = β 

L _~ ^(~ _n ^α+e;β) (x) − L _~ ^(~ _n ^α;β) (x)  and for i = 1, 2, . . . , r,

(3.2) L _~ ^(~ _n+e ^α;β) _i (x) = (βx + α i + 1)L ^(~ _{~ n} ^{α+e i ;β)} (x) + x d

dx L _~ ^(~ _n ^{α+e i ;β)} (x).

These equations can be regarded as a generalization of the relations d

dx L ^(α) _n (x) = L ^(α) _n (x) − L ^(α+1) _n (x) and

L ^(α) _n+1 (x) = (x − α − 1)L ^(α+1) _n (x) − x d

dx L ^(α+1) _n (x),

where {L ^(α) n (x)} ^∞ _n=0 is the monic Laguerre polynomials orthogonal with respect to w(x) = x ^α e ^−x on (0, ∞).

For a new recurrence relation for the multiple Laguerre I polynomials we use the identity

G ^{(~ α+e i ;β)} (x, t) = 1 1 − t i

G ^{(~ α;β)} (x, t) or equivalently

(3.3) G ^{(~ α;β)} (x, t) = (1 − t i )G ^{(~ α+e i ;β)} (x, t).

Theorem 3.1. Let {L _~ ^(~ _n ^α;β) (x)} ^∞ _{|~ n|=0} be the multiple Laguerre I polynomials.

Then we have for i = 1, 2, . . . , r,

(3.4) L ^(~ _{~ n} ^α;β) (x) = L ^(~ _{~ n} ^{α+e i ;β)} (x) − n i L ^(~ _{~ n−e} ^α+e _i ^{i ;β)} (x) and

(3.5) L ^(~ _{~ n} ^{α+e i ;β)} (x) =

n i

X

j=0

n i !

(n i − j)! L ^(α;β) _{~ n−je i} (x).

Proof. We prove here only the equation (3.5) because the equation (3.4) is an easy consequence of the identity (3.3). Since

∞

X

n i =0

∞

X

j=0

L ^(~ _{~ n} ^α;β) (x) t ⁿ _i ^{i +j} n i ! =

∞

X

n i =0 n i

X

j=0

L ^(~ _{~ n−je} ^α;β) _i (x) t ⁿ _i ⁱ

(n i − j)! ,

we have

1 1 − t i

G ^{(~ α;β)} (x, t) =





∞

X

j=0

t ^j _i





∞

X

~ n=0

L ^(~ _{~ n} ^α;β) (x) t ⁿ

~n!

!

= X _′





∞

X

n i =0

∞

X

j=0

L _~ ^(~ _n ^α;β) (x) t ⁿ _i ^{i +j} n i !





t ⁿ ₁ ¹ · · · t ⁿ _i−1 ⁱ⁻¹ t ⁿ _i+1 ⁱ⁺¹ · · · t ⁿ _r ^r n 1 ! · · · n i−1 !n i+1 ! · · · n r !

=

∞

X

~ n=0





n i

X

j=0

n i !

(n i − j)! L _~ ^(~ _n−je ^α;β) _i (x)



 t ^{~ n}

~n! , (3.6)

where P ′ := P ∞ n 1 =0

P ∞

n 2 =0 · · · P ∞ n i−1 =0

P ∞

n i+1 =0 · · · P ∞

n r =0 . Since (3.7) G ^{(~ α+e i ;β)} (x, t) =

∞

X

~ n=0

L _~ ^(~ _n ^{α+e i ;β)} (x) t ^{~ n}

~n! ,

we obtain the result by comparing the coefficients of (3.6) and (3.7).  Combining (3.2) and (3.4) we have

L ^(~ _{~ n+e} ^α;β) _i (x) − n i L ^(~ _{~ n} ^α;β) (x) = (βx + α i + 1)L ^(~ _{~ n} ^α;β) (x) + x d

dx L ^(~ _{~ n} ^α;β) (x), which can be obtained from (3.1) directly. In case of r = 2, Theorem 3.1 implies

L ^(α _{n 1} ¹ _,n ^,α ₂ ^{2 ;β)} (x) = L ^(α _{n 1} ¹ _,n ^+1,α ₂ ^{2 ;β)} (x) − n 1 L ^(α _{n 1} ¹ _−1,n ^+1,α ₂ ^{2 ;β)} (x)

= L ^(α _{n 1} ¹ _,n ^,α ₂ ^{2 +1;β)} (x) − n 2 L ^(α _{n 1} ¹ _,n ^,α ₂ ² ₋₁ ^+1;β) (x) and

L ^(α _{n 1} ¹ _,n ^+1,α ₂ ^{2 ;β)} (x) =

n 1

X

j=0

n 1 !

(n 1 − j)! L ^(α _{n 1} ¹ _−j,n ^{,α 2 ;β)} ₂ (x);

L ^(α _{n 1} ¹ _,n ^,α ₂ ^{2 +1;β)} (x) =

n 2

X

j=0

n 2 !

(n 2 − j)! L ^(α _{n 1} ¹ _,n ^,α ₂ ² _−j ^;β) (x).

If we adopt the relation (3.5) in Theorem 3.1 iteratively, we have L ^(~ _{~ n} ^α+e;β) (x) =

n 1

X

k 1 =0 n 2

X

k 2 =0

· · ·

n r

X

k r =0

~n!

(~n − ~k)! L ^(~ _{~ n−} ^{α;β) P} r

j=1 k j e j (x),

where ~k = (k 1 , k 2 , . . . , k r ). The recurrence relations in Theorem 3.1 can also be regraded as a generalization of

(3.8)

L ^(α) _n (x) = L ^(α+1) _n (x) + nL ^(α+1) _n−1 (x) and L ^(α+1) _n (x) =

n

X

j=0

(−1) ^j n!

(n − j)! L ^(α) _n−j (x),

where {L ^(α) n (x)} ^∞ _n=0 is the monic Laguerre polynomials.

4. Multiple Laguerre II polynomials

From the identities of G ^{(α;~ β)} (x, t) for multiple Laguerre II polynomials

∂

∂x G ^{(α;~ β)} (x, t) = |~ β · t|

1 − |t| G ^{(α;~ β)} (x, t) = |~ β · t|G ^{(α+1;~ β)} (x, t) and for i = 1, 2, . . . , r,

(4.1) ∂

∂t i

G ^{(α;~ β)} (x, t) = β i x + α + 1

1 − |t| G ^{(α;~ β)} (x, t) + x 1 − |t|

∂

∂x G ^{(α;~ β)} (x, t), the author found ([13, Theorem 2.8] differential-difference equations

d

dx L _~ ^(α;~ _n ^β) (x) =

r

X

j=1

β j n j L ^(α+1;~ _{~ n−e j} ^β) (x), d

dx L ^(α;~ _{~ n} ^β) (x) −

r

X

j=1

n j

d

dx L ^(α;~ _{~ n−e} ^β) _j (x) =

r

X

j=1

n j β j L ^(α;~ _{~ n−e} ^β) _j (x) and for i = 1, 2, . . . , r,

(4.2) L ^(α;~ _{~ n+e} ^β) _i (x) −

r

X

j=1

n j L ^(α;~ _{~ n−e} ^β) _{j +e i} (x) = (β i x + α + 1)L ^(α;~ _{~ n} ^β) (x) + x d

dx L ^(α;~ _{~ n} ^β) (x).

These equations can be regarded as generalizations of d

dx L ^(α) _n (x) = nL ^(α+1) _n−1 (x), d

dx

 L ^(α) _n (x) + nL ^(α) _n−1 (x) 

= nL ^(α) _n−1 (x) and

L ^(α) _n+1 (x) + nL ^(α) _n (x) = (x − α − 1)L ^(α) _n (x) − x d

dx L ^(α) _n (x), where {L ^(α) n (x)} ^∞ _n=0 is the monic Laguerre polynomials as in Section 3.

For a new recurrence relation for multiple Laguerre II polynomials we use the identity

G ^{(α+1;~ β)} (x, t) = 1

1 − |t| G ^{(α;~ β)} (x, t) or equivalently

G ^{(α;~ β)} (x, t) = (1 − |t|)G ^{(α+1;~ β)} (x, t).

Theorem 4.1. Let {L ^(α;~ _{~ n} ^β) (x)} ^∞ _{|~ n|=0} be the multiple Laguerre II polynomials.

Then we have

(4.3) L _~ ^(α;~ _n ^β) (x) = L ^(α+1;~ _{~ n} ^β) (x) −

r

X

j=1

n j L ^(α+1;~ _{~ n−e j} ^β) (x)

and

(4.4) L ^(α+γ+1;~ _{~ n} ^{β+~ δ)} (x) =

~ n

X

~ k=0

~n

~k

 L _~ ^{(γ;~ δ)}

k (x)L ^(α;~ _{~ n−} ^{β) P} r

j=1 k j e j (x), where γ > −1, ~δ = P r

i=1 δ i e i with δ i < 0, and ~k = (k 1 , k 2 , . . . , k r ).

Proof. From G ^{(α;~ β)} (x, t) = (1 − |t|)G ^{(α+1;~ β)} (x, t), the equation (4.3) can be proved by the equation

∞

X

~ n=0

L ^(α;~ _{~ n} ^β) (x) t ^{~ n}

~n! =

∞

X

~ n=0

(1 − |t|)L ^(α+1;~ _{~ n} ^β) (x) t ^{~ n}

~n!

=

∞

X

~ n=0

L ^(α+1;~ _{~ n} ^β) (x) t ^{~ n}

~n! −

∞

X

~ n=0

r

X

j=1

n j L _~ ^(α+1;~ _{n−e j} ^β) (x) t ^{~ n}

~n! . For the proof of the equation (4.4), note that

G ^{(α+γ+1;~ β+~ δ)} (x, t) = 1

(1 − |t|) ^α+γ+2 e ^{|(~ β+~ 1−|t| δ)·t|x} = G ^{(γ;~ δ)} (x, t)G ^{(α;~ β)} (x, t) so that

∞

X

~ n=0

L ^(α+γ+1;~ _{~ n} ^{β+~ δ)} (x) t ^{~ n}

~n! =





∞

X

~ k=0

L _~ ^{(γ;~ δ)}

k (x) t ^{~ k}

~k!





∞

X

~ n=0

L ^(α;~ _{~ n} ^β) (x) t ^{~ n}

~n!

!

=

r

X

i=1

∞

X

n i =0

∞

X

k i =0

L _~ ^{(γ;~ δ)}

k (x)L ^(α;~ _{~ n} ^β) (x) t ⁿ _i ^{i +k i} k i !n i !

!

=

r

X

i=1

∞

X

n i =0 n i

X

k i =0

n i

k i

 L _~ ^{(γ;~ δ)}

k (x)L ^(α;~ _{~ n−k} ^β) _{i e i} (x) t ⁿ _i ⁱ n i !

!

=

∞

X

~ n=0





~ n

X

~ k=0

~n

~k

 L _~ ^{(γ;~ δ)}

k (x)L ^(α;~ _{~ n−} ^{β) P} r

j=1 k j e j (x)



 t ^{~ n}

~n! . (4.5)

By comparing the coefficients of (4.5), the relation (4.4) follows.  Combining (4.2) and (4.3), we have

L ^(α−1;~ _{~ n+e i} ^β) (x) = (β i x + α + 1)L ^(α;~ _{~ n} ^β) (x) + x d

dx L ^(α;~ _{~ n} ^β) (x).

In case of r = 2, Theorem 4.1 implies

L ^(α;β _{n 1 ,n} ¹ ₂ ^{,β 2 )} (x) = L ^(α+1;β _{n 1 ,n 2} ^{1 ,β 2 )} (x) − n 1 L ^(α+1;β _{n 1 −1,n} ¹ ₂ ^{,β 2 )} (x) − n 2 L ^(α+1;β _{n 1 ,n 2 −1} ^{1 ,β 2 )} (x) and

L ^(α+γ+1;β _{n 1 ,n 2} ^{1 −1,β 2 −1)} (x) =

n 1

X

k 1 =0 n 2

X

k 2 =0

n 1

k 1

n 2

k 2



L ^{(γ;−1,−1)} _{k 1 ,k 2} (x)L ^(α;β _{n 1 −k} ^{1 ,β} _{1 ,n} ^{2 )} _{2 −k 2} (x).

The equation (4.3) is a generalization of the first equation (3.8) and the equation (4.4) is quite similar to

L ^(α+β+1) _n (x) =

n

X

k=0

n k



L ^(α) _k (0)L ^(β) _n−k (x).

5. Jacobi-Pi˜ neiro polynomials

For the generating function G ^{(α 1 ,α 2 ;α)} (x, t) of Jacobi-Pi˜ neiro polynomials, we can easily prove the identities

• t 2 G ^{(α 1 −1,α 2 ;α)} (x, t) − t 1 G ^{(α 1 ,α 2 −1;α)} (x, t) = (t 2 − t 1 )G ^{(α 1 ,α 2 ;α)} (x, t);

• t 1 xG ^{(α 1 +1,α 2 ;α)} (x, t) = (1 + t 1 )G ^{(α 1 ,α 2 −1;α)} (x, t) − G ^{(α 1 −1,α 2 −1;α)} (x, t);

t 2 xG ^{(α 1 ,α 2 +1;α)} (x, t) = (1 + t 2 )G ^{(α 1 −1,α 2 ;α)} (x, t) − G ^{(α 1 −1,α 2 −1;α)} (x, t);

• (x − 1)G ^{(α 1 ,α 2 ;α)} (x, t) = xG ^{(α 1 +1,α 2 +1;α−1)} (x, t) − G ^{(α 1 ,α 2 ;α−1)} (x, t), from which the following recurrence relations follow.

Theorem 5.1. Let {P n ^(α 1 ,n ^{1 ,α} 2 ^{2 ;α)} (x)} ^∞ _{n 1 +n 2 =0} be the Jacobi-Pi˜ neiro polynomials.

Then we have

(i) n 2 P _n ^(α _{1 ,n} ^{1 −1,α} _{2 −1} ^{2 ;α)} (x) − n 1 P _n ^(α _{1 −1,n} ^{1 ,α 2 −1;α)} ₂ (x)

= n 2 P _n ^(α _{1 ,n} ^{1 ,α} _{2 −1} ^{2 ;α)} (x) − n 1 P _n ^(α _{1 −1,n} ^{1 ,α 2 ;α)} ₂ (x);

(ii) n 1 xP _n ^(α _{1 −1,n} ^{1 +1,α} ₂ ^{2 ;α)} (x) = P n ^(α 1 ,n ^{1 ,α} 2 ^{2 −1;α)} (x) − P n ^(α 1 ,n ^{1 −1,α} 2 ^{2 −1;α)} (x) + n 1 P _n ^(α _{1 −1,n} ^{1 ,α 2 −1;α)} ₂ (x);

n 2 xP _n ^(α _{1 ,n} ^{1 ,α} _{2 −1} ^{2 +1;α)} (x) = P n ^(α 1 ,n ^{1 −1,α} 2 ^{2 ;α)} (x) − P n ^(α 1 ,n ^{1 −1,α} 2 ^{2 −1;α)} (x) + n 2 P _n ^(α _{1 ,n} ^{1 −1,α} _{2 −1} ^{2 ;α)} (x);

(iii) (x − 1)P n ^(α 1 ,n ^{1 ,α} 2 ^{2 ;α)} (x) = xP n ^(α 1 ,n ^{1 +1,α} 2 ^{2 +1;α−1)} (x) − P n ^(α 1 ,n ^{1 ,α} 2 ^{2 ;α−1)} (x).

Proof. From the first identity of generating function, we have

∞

X

n 1 =0

∞

X

n 2 =0

P _n ^(α _{1 ,n} ^{1 −1,α} ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ^{2 +1} n 1 !n 2 ! −

∞

X

n 1 =0

∞

X

n 2 =0

P _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 −1;α)} (x) t ⁿ ₁ ^{1 +1} t ⁿ ₂ ² n 1 !n 2 !

=

∞

X

n 1 =0

∞

X

n 2 =0

(t 2 − t 1 )P _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 !

so that (i) is proved. (ii) and (iii) can be proved by the same way.  Now, we will give a differential-difference relation for Jacobi-Pi˜ neiro poly- nomials. In order to obtain the relation, we need a differential equation of generating function.

Lemma 5.2. The generating function G ^{(α 1 ,α 2 ;α)} (x, t) of Jacobi-Pi˜ neiro poly- nomials satisfies

(5.1) x(x − 1)G x + [αx + (α 1 + 1)(x − 1)]G + t 1 (x − 1)G t 1 = G ^(α _{t 1} ^{1 ,α 2 ;α−1)}

and

(5.2) x(x − 1)G x + [αx + (α 2 + 1)(x − 1)]G + t 2 (x − 1)G t 2 = G ^(α _{t 2} ^{1 ,α 2 ;α−1)} , where G = G ^{(α 1 ,α 2 ;α)} (x, t), G x = ^dG _dx , G t 1 = _dt ^dG ₁ , and G t 2 = _dt ^dG ₂ .

Proof. For convenience we let A = 1 + t 1 − t 1 z, B = 1 + t 2 − t 2 z, C = (1 − t 1 z)(1 − t 2 z) − t 1 t 2 z, and Φ = AB − t 1 zB − t 2 zA. Then we have C = ^x−1 _z−1 ,

x = zAB and G ^{(α 1 ,α 2 ;α)} (x, t) = A ^{−α 1} B ^{−α 2} C ^−α Φ ⁻¹ . A direct calculation shows

G t 1 = −  α 1

A dA dt 1

+ α 2

B dB dt 1

+ α C

dC dt 1

+ 1 Φ

∂Φ

∂t 1

 G

= (z − 1)



α 1 (B − t 2 z) + α 2 t 2 z + αzB

z − 1 + (2z − 1)B

z − 1 − t 2 z − zB Φ

∂Φ

∂z

 G Φ and

G x = −  α 1

A dA dx + α 2

B dB

dx + α C

dC dx + 1

Φ

∂Φ

∂x

 G

= − α

x − 1 G +  α 1 t 1

A + α 2 t 2

B + α

z − 1 − 1 Φ

∂Φ

∂z

 G Φ . Hence,

(5.3)

xG x + αx

x − 1 G − A

z − 1 G t 1 = −



α 1 Φ + Φ + zAB

z − 1 + t 1 zB  G Φ

= −(α 1 + 1)G −  zAB

z − 1 + t 1 zB  G Φ . Since _∂t ^∂z ₁ = ^z(z−1)B _Φ , zB − zAB − t 1 zB(z − 1) = 0, and

(5.4)

A

z − 1 G t 1 = −t 1 G t 1 + 1 z − 1 G t 1

= −t 1 G t 1 + 1 x − 1

 x − 1 z − 1 G



t 1

+ zB

(z − 1)Φ G, we obtain by the equations (5.3) and (5.4)

xG x + αx

x − 1 G + t 1 G t 1 + (α 1 + 1)G = 1

x − 1 G ^(α _{t 1} ^{1 ,α 2 ,α−1)}

which is (5.1). The equation (5.2) can be proved by the same method.  Theorem 5.3. The Jacobi-Pi˜ neiro polynomial {P n ^(α 1 ,n ^{1 ,α} 2 ^{2 ;α)} (x)} ^∞ _{n 1 +n 2 =0} satisfies a differential-difference equation

(5.5)

P _n ^(α

₁

_+1,n

¹

^,α

²

^;α−1)

₂

(x) = x(x − 1) d

dx P _n ^(α

₁

_,n

¹

^,α

₂²

^;α) (x) + [αx + (α 1 + 1 + n 1 )(x − 1)]P _n ^(α

₁

_,n

¹

^,α

₂²

^;α) (x)

and (5.6)

P _n ^(α

₁

_,n

¹

^,α

₂

₊₁

²

^;α−1) (x) = x(x − 1) d

dx P _n ^(α

₁

_,n

¹

^,α

₂²

^;α) (x) + [αx + (α 2 + 1 + n 2 )(x − 1)]P _n ^(α

₁

_,n

¹

^,α

₂²

^;α) (x) . Proof. Since

t 1 G t 1 =

∞

X

n 1 =0

∞

X

n 2 =0

n 1 P _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! , G t 1 =

∞

X

n 1 =0

∞

X

n 2 =0

P _n ^(α _{1 +1,n} ^{1 ,α 2 ;α)} ₂ (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! , G x =

∞

X

n 1 =0

∞

X

n 2 =0

d

dx P _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! ,

we obtain the equation (5.5) from the equation (5.1) in Lemma 5.2. The second equation (5.6) can also be proved by the same method using the equation (5.2)

in Lemma 5.2. 

The equations (5.5) and (5.6)) are kinds of raising operators that are very useful to find a differential equation for orthogonal polynomials. It is well known that the Jacobi polynomial {P n ^(α,β) (x)} ^∞ _n=0 satisfies a differential-difference equation

(1 − x ² ) d

dx P _n ^(α,β) (x) = n + α + β + 1

2n + α + β + 2 [(2n + α + β + 2)x + α − β]P _n ^(α,β) (x)

− 2(n + 1)(n + α + β + 1)

2n + α + β + 2 P _n+1 ^(α,β) (x),

where {P n ^(α,β) (x)} ^∞ _n=0 is orthogonal with respect to w(x) = (1 − x) ^α (1 + x) ^β on [−1, 1] and normalized by P n ^(α,β) (1) = ^n+α _n
. Hence, the results of Theorem 5.3 is a generalization of the relation for classical Jacobi polynomials.

Subtracting (5.6) from (5.5) in Theorem 5.3 gives

(α 1 − α 2 + n 1 − n 2 )(x − 1)P _n ^(α _{1 ,n} ^{1 ,α} ₂ ^{2 ;α)} (x) = P _n ^(α _{1 +1,n} ^{1 ,α 2 ;α−1)} ₂ (x) − P _n ^(α _{1 ,n} ^{1 ,α} _{2 +1} ^{2 ;α−1)} (x) and subtracting the equation (5.6) of the case (n 1 , n 2 − 1) from the equation (5.5) of the case (n 1 − 1, n 2 ), we have

x(x − 1) d dx



P _n ^(α _{1 −1,n} ^{1 ,α 2 ;α)} ₂ (x) − P _n ^(α _{1 ,n} ^{1 ,α} _{2 −1} ^{2 ;α)} (x) 

= αx 

P _n ^(α _{1 ,n} ^{1 ,α} _{2 −1} ^{2 ;α)} (x) − P _n ^(α _{1 −1,n} ^{1 ,α 2 ;α)} ₂ (x)  + (x − 1) 

(α 2 + n 2 )P _n ^(α _{1 ,n} ^{1 ,α} _{2 −1} ^{2 ;α)} (x) − (α 1 + n 1 )P _n ^(α _{1 −1,n} ^{1 ,α 2 ;α)} ₂ (x) 

,

which seems to be a new differential-difference equation for Jacobi-Pi˜ neiro poly-

nomials.

6. Multiple Bessel polynomials

For the generating function G ^{(α 1 ,α 2 ;γ)} (x, t) of multiple Bessel polynomials, we can easily prove the identities

• t 2 G ^{(α 1 −1,α 2 ;γ)} (x, t) − t 1 G ^{(α 1 ,α 2 −1;γ)} (x, t) = (t 2 − t 1 )G ^{(α 1 ,α 2 ;γ)} (x, t).

• t 1 xG ^{(α 1 +1,α 2 ;γ)} (x, t) = G ^{(α 1 ,α 2 −1;γ)} (x, t) − G ^{(α 1 −1,α 2 −1;γ)} (x, t);

t 2 xG ^{(α 1 ,α 2 +1;γ)} (x, t) = G ^{(α 1 −1,α 2 ;γ)} (x, t) − G ^{(α 1 −1,α 2 −1;γ)} (x, t) from which the following recurrence relations follow.

Theorem 6.1. Let {B ^(α n 1 ¹ ,n ^,α 2 ^{2 ;γ)} (x)} ^∞ _{n 1 +n 2 =0} be the multiple Bessel polynomials.

Then we have

(i) n 2 B _n ^(α ₁ ¹ _,n ^−1,α _{2 −1} ^{2 ;γ)} (x) − n 1 B _n ^(α ₁ ¹ _−1,n ^{,α 2 −1;γ)} ₂ (x)

= n 2 B _n ^(α ₁ ¹ _,n ^,α _{2 −1} ^{2 ;γ)} (x) − n 1 B ^(α _{n 1} ¹ _−1,n ^{,α 2 ;γ)} ₂ (x).

(ii) n 1 xB _n ^(α ₁ ¹ _−1,n ^+1,α ₂ ^{2 ;γ)} (x) = B n ^(α 1 ¹ ,n ^,α 2 ^{2 −1;γ)} (x) − B n ^(α 1 ¹ ,n ^−1,α 2 ^{2 −1;γ)} (x);

n 2 xB _n ^(α ₁ ¹ _,n ^,α ₂ ² ₋₁ ^+1;γ) (x) = B n ^(α 1 ¹ ,n ^−1,α 2 ^{2 ;γ)} (x) − B n ^(α 1 ¹ ,n ^−1,α 2 ^{2 −1;γ)} (x).

Proof. From the first equation of generating function, we have

∞

X

n 1 =0

∞

X

n 2 =0

B ^(α _{n 1} ¹ _,n ^−1,α ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ^{2 +1} n 1 !n 2 ! −

∞

X

n 1 =0

∞

X

n 2 =0

B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 −1;α)} (x) t ⁿ ₁ ^{1 +1} t ⁿ ₂ ² n 1 !n 2 !

=

∞

X

n 1 =0

∞

X

n 2 =0

(t 2 − t 1 )B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 ;α)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 !

so that (i) is proved. (ii) and (iii) can be proved by the same way.  Subtracting the second equation from the first equation in Theorem 6.1(ii), we obtain an interesting relation

B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 −1;γ)} (x)−B _n ^(α ₁ ¹ _,n ^−1,α ₂ ^{2 ;γ)} (x) = x 

n 1 B _n ^(α ₁ ¹ _−1,n ^+1,α ₂ ^{2 ;γ)} (x) − n 2 B _n ^(α ₁ ¹ _,n ^,α ₂ ² ₋₁ ^+1;γ) (x)  . Now, we will give a differential-difference relation for multiple Bessel poly- nomials. In order to obtain the relation, we need a differential equation of generating function.

Lemma 6.2. The generating function G ^{(α 1 ,α 2 ;γ)} (x, t) of multiple Bessel poly- nomials satisfies

(6.1) x ² G x + [(α 1 + 1)x − γ]G + t 1 xG t 1 = G ^(α _{t 1} ^{1 −1,α 2 −1;γ)} and

(6.2) x ² G x + [(α 2 + 1)x − γ]G + t 2 xG t 2 = G ^(α _{t 2} ^{1 −1,α 2 −1;γ)} , where G = G ^{(α 1 ,α 2 ;γ)} (x, t), G x = ^dG _dx , G t 1 = ^dG _{dt 1} , and G t 2 = _dt ^dG ₂ .

Proof. For convenience, we let A = 1 − t 1 z, B = 1 − t 2 z, and Φ = AB − t 1 zB − t 2 zA. Then we have

x = zAB and G ^{(α 1 ,α 2 ;α)} (x, t) = A ^{−α 1} B ^{−α 2} e ^{γ( z 1 − 1 x )} .

A direct calculation similar to the proof of Lemma 6.2 shows G t 1 =



α 1 z(B − t 2 z) + α 2 t 2 z ² − γB + 2zB − t 2 z ² − z ² B Φ

∂Φ

∂z

 G Φ and

G x = γ

x ² G +  α 1 t 1

A + α 2 t 2

B − γ z ² − 1

Φ

∂Φ

∂z

 G Φ so that

xG x − γ x G − A

z G t 1 = −α 1 G + (t 2 zA − 2AB) G Φ . Hence, we have

xG x − γ

x G + t 1 G t 1 + α 1 G = 1

z G t 1 + (t 2 zA − 2AB) G Φ

= 1

x (ABG) t 1 + (B + t 2 zA − 2AB) G Φ

= 1

x G ^(α _{t 1} ^{1 −1,β 1 −1;γ)} − G

which is (6.1). The equation (6.2) can be proved by the same method.  Theorem 6.3. The multiple Bessel polynomial {B n ^(α 1 ¹ ,n ^,α 2 ^{2 ;γ)} (x)} ^∞ _{n 1 +n 2 =0} satis- fies a differential-difference equation

(6.3) x ² d

dx B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 ;γ)} (x)+[(α 1 +1+n 1 )x−γ]B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 ;γ)} (x) = B ^(α _{n 1} ¹ _+1,n ^−1,α ₂ ^{2 −1;γ)} (x) and

(6.4) x ² d

dx B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 ;γ)} (x)+[(α 2 +1+n 2 )x−γ]B _n ^(α ₁ ¹ _,n ^,α ₂ ^{2 ;γ)} (x) = B _n ^(α ₁ ¹ _,n ^−1,α _{2 +1} ^{2 −1;γ)} (x).

Proof. Since

t 1 G t 1 =

∞

X

n 1 =0

∞

X

n 2 =0

n 1 B ^(α _{n 1} ¹ _,n ^,α ₂ ^{2 ;γ)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! , G t 1 =

∞

X

n 1 =0

∞

X

n 2 =0

B ^(α _{n 1} ¹ _+1,n ^{,α 2 ;γ)} ₂ (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! , G x =

∞

X

n 1 =0

∞

X

n 2 =0

d

dx B ^(α _{n 1} ¹ _,n ^,α ₂ ^{2 ;γ)} (x) t ⁿ ₁ ¹ t ⁿ ₂ ² n 1 !n 2 ! ,

we prove the equation (6.3) from the equation (6.1). The equation (6.4) can

be proved by the same method using the equation (6.2) in Lemma 6.2. 

The equations (6.3) and (6.4) are kinds of raising operators for multiple

Bessel polynomials that is very useful to find a differential equation for or-

thogonal polynomials. It is well known that the monic Bessel polynomial

{B n ^(α,β) (x)} ^∞ _n=0 satisfies a differential-difference equation x ² d

dx B _n ^(α,β) (x) =



nx − βn

2n + α − 2



B _n ^(α,β) (x) + n(n + α − 2)β ²

(2n + α − 3)(2n + α − 2) ² B ^(α,β) _n−1 (x),

where {B n ^(α,β) (x)} ^∞ _n=0 is orthogonal with respect to w(x) = x ^α e ^{β x} on the unit circle in complex plane. Hence, the result of Theorem 6.3 is a generalization of the relation for classical Bessel polynomials.

Subtracting the equation (6.4) from (6.3) in Theorem 6.3 gives

(α 1 − α 2 + n 1 − n 2 )xB ^(α _{n 1} ¹ _,n ^,α ₂ ^{2 ;γ)} (x) = B _n ^(α ₁ ¹ _+1,n ^−1,α ₂ ^{2 −1;α)} (x) − B _n ^(α ₁ ¹ _,n ^−1,α _{2 +1} ^{2 −1;α)} (x) and subtracting the equation (6.4) of the case (n 1 , n 2 − 1) from (6.3) of the case (n 1 − 1, n 2 ), we get

x ² d dx

 B _n ^(α ₁ ¹ _−1,n ^{,α 2 ;γ)} ₂ (x) − B _n ^(α ₁ ¹ _,n ^,α _{2 −1} ^{2 ;γ)} (x) 

= γ 

B _n ^(α ₁ ¹ _−1,n ^{,α 2 ;γ)} ₂ (x) − B _n ^(α ₁ ¹ _,n ^,α _{2 −1} ^{2 ;γ)} (x) 

− x 

(α 1 + n 1 )B ^(α _{n 1} ¹ _−1,n ^{,α 2 ;γ)} ₂ (x) − (α 2 + n 2 )B _n ^(α ₁ ¹ _,n ^,α _{2 −1} ^{2 ;γ)} (x)  ,

which seems to be a new differential-difference equation for multiple Bessel polynomials.

Acknowledgements. The author thanks the referee for his/her careful read- ing to improve this paper. This research was supported by Kyungpook National University Research Fund, 2012.

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Comput. Appl. Math. 127 (2001), no. 1-2, 317–347.

Department of Mathematics Teachers College

Kyungpook National University Daegu 702-701, Korea

E-mail address: [email protected]