The aim of this work is twofold, on the one hand the associated qorthogonal polynomials with a class of discrete q-distributions, by their weight functions are derived and on the other hand the combinatorial interpretation of these q-orthogonal polynomials is presented. Speci cally, 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, a ne 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 presented by the authors.
Kemp [Kem92a, Kem92b], introduced Heine and Euler, q-Poisson distributions, with probability functions given respectively by fXH (x) = eq( ; x = 0; 1; 2; : : : ; 0 < q < 1; 0 < < 1 and where and fXE (x) = Eq( ; x = 0; 1; 2; : : : ; 0 < q < 1; 0 < (1
q) < 1; eq(z) :=
X1 (1 n=0
q)nzn (q; q)n
1 = X
zn n=0 [n]q! ((1 1 q)z; q)1 ; jzj < 1 Eq(z) :=
X1 (1 n=0
z q)nq(2)zn (q; q)n
z X1 q(2)zn n=0 [n]q! = ((1 q)z; q)1; jzj < 1: [x]q! )
x [x]q! (1) (2) (3) (4) Charalambides [Cha10, Cha16], derived Heine as direct approximation, as n ! 1, of the q-Binomial I and the q-negative Binomial I, with probability functions given respectively by and and fXB(x) = n x q j=1 n q(x2) x Y(1 + qj 1) 1; x = 0; 1; : : : ; n; fXNB(x) = n + x x 1 q
n+x q(x2) x Y (1 + qj 1) 1; x = 0; 1; : : : ;
j=1 fXBS (x) = n x q
n x x Y (1 j=1
qj 1); x = 0; 1; : : : ; n; fXNBS (x) = n + x x 1 q
n x Y(1 j=1 qj 1); x = 0; 1; : : : ; where
> 0, 0 < q < 1:
Moreover, Charalambides [Cha10, Cha16], derived Euler distribution as direct approximation, as n ! 1, of the q-Binomial II and the negative q-Binomial II, with probability functions given respectively by where 0 < < 1 and 0 < q < 1 or 1 < q < 1 with qn 1 < 1: Kyriakoussis and Vamvakari [KV10] introduced deformed types of the q-negative Binomial of the rst kind, q-binomial of the rst 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 de ned 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, a ne 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. 2
Let v be a probability measure in R with nite moments of all orders
Z
R sm =
xmdv(x): (5) (6) (7) (8) Then there exist a sequence of normalized orthogonal polynomials fpm(x)g with respect to the measure v satisfying the recurrence relation xpm(x) = pm+1(x) + ampm(x) + bmpm 1(x); m 1; (9) with initial conditions Moreover, they satisfy the orthogonality relation where m the Kronecker delta.
The polynomials fpm(x)g depend on the moment sequence fsmgm 0 and they can be obtain from the formula pm(x) = s b1b2
bm Dm 1Dm s0 s1 : : : sm s1 s2 : : : sm+1 : : : : : : : : : : : : sm 1 sm : : : s2m 1 1 x : : : xm ; where Dm = det(fsi+jg0 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 nitely many points if and only if bm = 0 for some m on, thus the sequence of
polynomials is essentially nite. The mean value and the variance of the random variable X in R with probability
density function v(x) are given respectively by
(
2 = b1:
Note that the probability measure v is uniquely determined if the coe cinets am and bm in the reccurence relation (9) are bounded when m ! 1 (see Christiansen [Chr04]).
The q-orthogonal polynomials Little q-Jacobi, a ne q-Krawtchouk and little q-Laguerre/Wall satisfy the recurrence relation (9) with am and bm given in the next table respectively. fYBS (y) =
n g(y) q g(y) n g(y)
Y (1 j=1 y = [0]q; [1]q; [2]q; : : : ; qj 1); 0 <
< 1; 0 < q < 1;
3.1 where
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 de ned in the spectrum S = f[x]q; x = 0; 1; : : :g with p.f.
f NBS (y) Y = n + g(y) g(y) 1 q
n g(y) Y(1
j=1 y = [0]q; [1]q; [2]q; : : : ; qj 1); 0 <
< 1; 0 < q < 1; 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 NBS (y) is induced by the normalized linear transforY mation of the little q-Jacobi orthogonal polynomials, say pJm(y; a; b; q), 0 < q < 1; given by pJm(y; a; b; q) = ( 1)m(abqm+1; q)m pLmitJ (y; a; b; q); q( m2)(1 q)m(aq; q)m where y = [x]q; x = 0; 1; : : : ; and pLmitJ (y; a; b; q) the little q-Jacobi orthogonal polynomials with parameter a = =q and b = qn 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 de ned in the spectrum S = f[x]q; x = 0; 1; : : :g with p.f. (13) (14) (15) : Using the orthogonality relation of the normalized a ne 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) is induced by the normalized linear transformaY tion of deformed a ne q-Krawtchouk orthogonal polynomials, say pAmK (y; p; q; n), 0 < q < 1; given by pAmK (y; p; n; q) = (1
q) m(pq; q n; q)mpAmff (q ny; p; n; q); where y = [x]q; x = 0; 1; : : : ; and pAmff (q ny; p; n; q) the deformed a ne q-Krawtchouk orthogonal polynomials with parameter p = =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 de ned in the spectrum S = f[x]q;
x = 0; 1; : : :g with p.f.
where
where
(
Remark 3.4. The approximation, as n ! 1; 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 ! 1; 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 a ne q-Krawtchouk to the little q-Laguerre/Wall ones. The above mentioned conclusions can be justi ed since the coe cients in the recurrence relation of the associated q-orthogonal polynomials are bounded in m. 3.2
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 wi, 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 WG(M ), is de ned by fYE (y) =
Eq( )
g(y) [g(y)]q! y = and the corresponding generating function in m variables including the vertex and edge weights is de ned by P (G; w1; w2; : : : ; wm) =
X( 1)jMj M
Y where jM j the number of edges in M , have proved the following proposition.
Proposition 3.5. The generating function of matching sets in paths, Lm; satis es the recurrence relation Lm+1 = wn+1Lm
W (m; m + 1)Lm 1; m = 0; 1; 2; : : : with initial conditions L 1 = 0 and L0 = 1.
Remark 3.6. Setting in (20) vertex weight wm = x am and edge weight W (m; m + 1) = bm, where am and bm are bounded sequences in m, and comparing (20) with (9), we have a wide class of generating functions of matching sets in paths identi ed with q-orthogonal polynomials. Between them the little q-Jacobi, a ne q-Krawtchouk and little q-Laguerre/Wall polynomials where the bounded sequences am and bm are given respectively in the Table 1.
Remark 3.7. 3. Setting in (20) vertex weight wm = [x]q dm and edge weight W (m; m + 1) = gm, where dm and gm are bounded sequences in m and comparing (20) with (9) we have a wide class of generating functions of matching sets in paths identi ed with transformed q-orthogonal polynomials. Between them the associated transformed little q-Jacobi, a ne 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. (19) (20) [HL72]