Introduction

Kemp [Kem92a,Kem92b], introduced Heine and Euler, q-Poisson distributions, with probability functions given respectively by f H X (x) = e q (−λ) q ( x 2 ) λ x [x] q ! , x = 0, 1, 2, . . . , 0 < q < 1, 0 < λ < ∞ (1)

and

f E X (x) = E q (−λ) λ x [x] q ! , x = 0, 1, 2, . . . , 0 < q < 1, 0 < λ(1 − q) < 1,(2)

where

e q (z) := ∞ n=0 (1 − q) n z n (q; q) n = ∞ n=0 z n [n] q ! = 1 ((1 − q)z; q) ∞ , |z| < 1 (3)

and

E q (z) := ∞ n=0 (1 − q) n q ( z 2 ) z n (q; q) n = ∞ n=0 q ( z 2 ) z n [n] q ! = ((1 − q)z; q) ∞ , |z| < 1.(4)

Charalambides [Cha10,Cha16], derived Heine as direct approximation, as n → ∞, of the q-Binomial I and the q-negative Binomial I, with probability functions given respectively by

f B X (x) = n x q q ( x 2 ) θ x n j=1

(1 + θq j−1 ) −1 , x = 0, 1, . . . , n,

and

f N B X (x) = n + x − 1 x q q ( x 2 ) θ x n+x j=1

(1 + θq j−1 ) −1 , x = 0, 1, . . . ,

where θ > 0, 0 < q < 1.

Moreover, Charalambides [Cha10,Cha16], derived Euler distribution as direct approximation, as n → ∞, of the q-Binomial II and the negative q-Binomial II, with probability functions given respectively by

f BS X (x) = n x q θ x n−x j=1 (1 − θq j−1 ), x = 0, 1, . . . , n,(7)

and

f N BS X (x) = n + x − 1 x q θ x n j=1 (1 − θq j−1 ), x = 0, 1, . . . ,(8)

where 0 < θ < 1 and 0 < q < 1 or 1 < q < ∞ with θq n−1 < 1. Kyriakoussis and Vamvakari [KV10] introduced deformed types of the q-negative Binomial of the first kind, q-binomial of the first kind and of the Heine distributions and derived the associated q-orthogonal polynomials, based on discrete q-Meixner, q-Krawtchouk and q-Charlier orthogonal polynomials respectively. Moreover, Kyriakoussis and Vamvakari [KV12] established families of terminating and non-terminating q-Gauss hypergeometric series discrete distributions and associated them with defined classes of generalized q-Hahn and big q-Jacobi orthogonal polynomials, respectively. Also, Kyriakoussis and Vamvakari [KV05] presented generalization of matching extensions in graphs and provided combinatorial interpretation of wide classes of orthogonal and q-orthogonal polynomials as generating functions of matching sets in paths.

In this paper, we derive the associated q-orthogonal polynomials with some deformed types of the q-negative Binomial of the second kind, q-binomial of the second kind and Euler distributions. The derived q-orthogonal polynomials are based on the little q-Jacobi, affine q-Krawtchouk and little q-Laguerre/Wall orthogonal polynomials respectively. Also, we provide a combinatorial interpretation of these q-orthogonal polynomials, as applications of a generalization of matching extensions in paths, already prresented by the authors.

For the needs of this paper the class of discrete q-distributions, q-negative Binomial I, q-Binomial I and Heine will be called class of discrete q-distributions of type I, while the class of discrete q-distributions, q-negative Binomial II, q-Binomial II and Euler will be called class of discrete q-distributions of type II.

Preliminaries

Let v be a probability measure in R with finite moments of all orders

s m = R x m dv(x).

Then there exist a sequence of normalized orthogonal polynomials {p m (x)} with respect to the measure v satisfying the recurrence relation

xp m (x) = p m+1 (x) + a m p m (x) + b m p m−1 (x), m ≥ 1,(9)

with initial conditions

p 0 (x) = 1, p 1 (x) = x − a 0 .

Moreover, they satisfy the orthogonality relation

S p m (x)p ν (x)dv(x) = δ mν b 1 b 2 • • • b m , m, ν ≥ 0 (10)

where δ mν the Kronecker delta.

The polynomials {p m (x)} depend on the moment sequence {s m } m≥0 and they can be obtain from the formula

p m (x) = b 1 b 2 • • • b m D m−1 D m s 0 s 1 . . . s m s 1 s 2 . . . s m+1 . . . . . . . . . . . . s m−1 s m . . . s 2m−1 1 x . . . x m ,(11)

where D m = det({s i+j } 0≤i,j≤m ) denotes the Hankel determinant.

Conversely, Favard's (1935) theorem ensures the existence of a probability measure v on R for which the sequence of polynomials determined by the recurrence relation ( 9) are orthogonal. It can also be shown that the probability measure v is supported only in finitely many points if and only if b m = 0 for some m on, thus the sequence of polynomials is essentially finite. The mean value and the variance of the random variable X in R with probability density function v(x) are given respectively by µ = a 0 and σ 2 = b 1 .

If a m = 0 then all moments of odd order are zero

s 2m+1 = x∈R x 2m+1 dv(x) = 0.

Also, from the recurrence relation (9) the following representation of orthogonal polynomials is derived

p m (x) = p 0 (x) x + a 1 b 1/2 2 0 . . . . . . 0 b 1/2 2 x + a 2 b 1/2 3 0 . . . 0 b 1/2 3 x + a 3 . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . x + b m−1 c 1/2 m 0 . . . . . . 0 b 1/2 m x + b m .(12)

(see Szegö( [Sze59], p.374).

Note that the probability measure v is uniquely determined if the coefficinets a m and b m in the reccurence relation (9) are bounded when m → ∞ (see Christiansen [Chr04]).

The q-orthogonal polynomials Little q-Jacobi, affine q-Krawtchouk and little q-Laguerre/Wall satisfy the recurrence relation (9) with a m and b m given in the next table respectively.

Table 1 : q-Classical Orthogonal Polynomials: Little q-Jacobi, affine q-Krawtchouk and little q-Laguerre/Wall

Little q-Jacobi p LitJ m (x; a, b; q) a m q m (1+a 2 bq m+1 +a(1−(1+b)q m −(1+b)q m −(1+b)q m+2 +bq 2m+1 ))

(1−abq 2m )(1−abq 2m+2 ) b m −aq m+1 (1−q m )(1−aq m )(1−bq m )(1−abq m )(c−abq m )(1−cq m ) (1−abq 2m ) 2 (1−abq 2m−1 )(1−abq 2m+1 ) Affine q-Krawtchouk p Af f m (x; p, n, q) a m 1 − (1 − q m−n )(1 − pq m+1 ) − pq m−n (1 − q m ) b m pq m−n (1 − q m )(1 − pq m )(1 − q m−n−1 ) Little q-Laguerre/Wall p LLW m (x, a; q) a m q m (1 − aq m+1 ) + aq m (1 − q m ) b m aq 2m−1 (1 − q m )(1 − aq m )

3 Main Results

3.1

Associated q-Orthogonal Polynomials with the class of Discrete q-Distributions of type II

In this section we derive the associated q-orthogonal polynomials with the class of discrete q-distributions of type II, (8),( 7) and (2), in respect to their weight functions. We begin by transfering from the random variable X of the q-negative Binomial of the second kind distribution (8) to the equal-distributed deformed random variable Y = [X] q , and we obtain a deformed q-negative Binomial II distribution defined in the spectrum S = {[x] q , x = 0, 1, . . .} with p.f.

f N BS Y (y) = n + g(y) − 1 g(y) q θ g(y) n j=1 (1 − θq j−1 ), 0 < θ < 1, 0 < q < 1, y = [0] q , [1] q , [2] q , . . . ,(13)

where g(y) = ln(1 − (1 − q)y) ln q .

Using the orthogonality relation of the normalized little q-Jacobi orthogonal polynomials [Ism05, KS98] and the linear transformation of orthogonal polynomials [Sze59], we easily derive the following result.

Proposition 3.1. The probability distribution with p.f. f N BS Y (y) is induced by the normalized linear transformation of the little q-Jacobi orthogonal polynomials, say p J m (y; a, b, q), 0 < q < 1, given by

p J m (y; a, b, q) = (−1) m (abq m+1 ; q) m q ( m 2 ) (1 − q) m (aq; q) m p LitJ m (y; a, b, q),(14)

where y = [x] q , x = 0, 1, . . . , and p LitJ m (y; a, b, q) the little q-Jacobi orthogonal polynomials with parameter a = θ/q and b = q n−1 .

Next, we transfer from the random variable X of the q-Binomial of the second kind distribution (7) to the equal-distributed deformed random variable Y = [X] q , and we obtain a deformed q-Binomial II distribution defined in the spectrum S = {[x] q , x = 0, 1, . . .} with p.f.

f BS Y (y) = n g(y) q θ g(y)

n−g(y)

j=1

(1 − θq j−1 ), 0 < θ < 1, 0 < q < 1,

y = [0] q , [1] q , [2] q , . . . ,(15)

where

g(y) = ln(1 − (1 − q)y) ln q .

Using the orthogonality relation of the normalized affine q-Krawtchouk orthogonal polynomials [Ism05,KS98] and the linear transformation of orthogonal polynomials [Sze59], we easily derive the following result.

Proposition 3.2. The probability distribution with p.f. f BS Y (y) is induced by the normalized linear transformation of deformed affine q-Krawtchouk orthogonal polynomials, say p AK m (y; p, q, n), 0 < q < 1, given by p AK m (y; p, n, q) = (1 − q) −m (pq, q −n ; q) m p Af f m (q −n y; p, n, q), (16

)

where y = [x] q , x = 0, 1, . . . , and p Af f m (q −n y; p, n, q) the deformed affine q-Krawtchouk orthogonal polynomials with parameter p = θ/q.

Finally, we transfer from the random variable X of the Euler distribution (2) to the equal-distributed deformed random variable Y = [X] q , and we obtain a deformed Euler distribution defined in the spectrum S = {[x] q , x = 0, 1, . . .} with p.f.

f E Y (y) = E q (−λ) λ g(y) [g(y)] q ! , 0 < q < 1, 0 < λ(1 − q) < 1, y = [0] q , [1] q , [2] q , . . . ,(17)

where

g(y) = ln(1 − (1 − q)y) ln q .

Using the orthogonality relation of the normalized little q-Laguerre/Wall orthogonal polynomials [Ism05,KS98] and the linear transformation of orthogonal polynomials [Sze59], we easily derive the following result.

Proposition 3.3. The probability distribution with p.f. f E Y (y) is induced by the normalized linear transformation of the little q-Laguerre/Wall orthogonal polynomials, say p L m (y; a, q), 0 < q < 1, given by

p L m (y; a, q) = (−1) m (aq; q) m (1 − q) −m q ( m 2 ) p LLW m (y; a, q),(18)

where y = [x] q , x = 0, 1, . . . , and p LLW m (y; a, q) the little q-Laguerre/Wall orthogonal polynomials with parameter a = λ(1 − q). Remark 3.4. The approximation, as n → ∞, of the q-Binomial I and the q-negative Binomial I to the Heine distribution, can alternatively be concluded by the limit of the associated q-orthogonal polynomials, q-Krawtchouk and q-Meixner to the q-Charlier ones. Also, the approximation, as n → ∞, of the q-Binomial II and the qnegative Binomial II to the Euler distribution, can also be concluded by the limit of the associated little q-Jacobi and affine q-Krawtchouk to the little q-Laguerre/Wall ones. The above mentioned conclusions can be justified since the coefficients in the recurrence relation of the associated q-orthogonal polynomials are bounded in m.

Combinatorial Interpetation of the Associated q-Orthogonal Polynomials

Combinatorial interpretation of orthogonal polynomials using matchings in graphs has received much attention by several authors over the last decades. Among them we refer to Feinsilver et al [FSS96], Godsil [God81], Godsil and Gutman [GG81], Viennot [Vie83], and Heilmann and Lieb [HL72], Kyriakoussis and Vamvakari [KV05]. Let G be a simple graph on m vertices with vertex labels 1 to m, having edge weight W (i, j) a non-negative real number for each unordered pair of vertices < i, j > , i = 1, 2, . . . , m, j = 1, 2, . . . , m, i < j and vertex weight w i , i = 1, 2, . . . , m. Also, let M be a matching set of G consisting of disjoint edges pairwise having no vertex in common. Then the weight of M , say W G (M ), is defined by

W G (M ) = i,j M W (i, j) i ∈M w i

and the corresponding generating function in m variables including the vertex and edge weights is defined by

P (G; w 1 , w 2 , . . . , w m ) = M (−1) |M | i,j M W (i, j) i ∈M w i

with |M | the number of edges in M , summing over all matchings M of G . Let L m be a path on m vertices with edge weight W (i, j) > 0 when |i − j| = 1, W (i, j) = 0 otherwise and with vertex weight w i , i = 1, 2, . . . , m. Note that w i and W (i, i + 1) are bounded sequences in i, i = 1, 2, . . . . Kyriakoussis and Vamvakari [KV05], setting

L n = P (L m ; w 1 , w 2 , . . . , w m ) = M (−1) |M | i,j M W (i, j) i ∈M w i(19)

where |M | the number of edges in M , have proved the following proposition. 9), we have a wide class of generating functions of matching sets in paths identified with q-orthogonal polynomials. Between them the little q-Jacobi, affine q-Krawtchouk and little q-Laguerre/Wall polynomials where the bounded sequences a m and b m are given respectively in the Table 1. 9) we have a wide class of generating functions of matching sets in paths identified with transformed q-orthogonal polynomials. Between them the associated transformed little q-Jacobi, affine q-Krawtchouk, and little q-Laguerre/Wall polynomials with the deformed q-negative binomial of type II, q-binomial of type II and Euler distributions respectively.