On associated q-orthogonal polynomials with a class of discrete q-distributions Andreas Kyriakoussis Malvina Vamvakari akyriak@hua.gr mvamv@hua.gr Harokopion University, Department of Informatics and Telematics, Athens, Greece Abstract The aim of this work is twofold, on the one hand the associated q- orthogonal 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. Specifi- cally, we derive the associated q-orthogonal polynomials with some de- formed 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 polynomi- als, as applications of a generalization of matching extensions in paths, already presented by the authors. 1 Introduction Kemp [Kem92a, Kem92b], introduced Heine and Euler, q-Poisson distributions, with probability functions given respectively by x H q ( 2 ) λx fX (x) = eq (−λ) , x = 0, 1, 2, . . . , 0 < q < 1, 0 < λ < ∞ (1) [x]q ! and E λx fX (x) = Eq (−λ) , x = 0, 1, 2, . . . , 0 < q < 1, 0 < λ(1 − q) < 1, (2) [x]q ! where ∞ ∞ X (1 − q)n z n X zn 1 eq (z) := = = , |z| < 1 (3) n=0 (q; q)n n=0 [n] q ! ((1 − q)z; q)∞ and ∞ z ∞ z X (1 − q)n q (2) z n X q (2) z n Eq (z) := = = ((1 − q)z; q)∞ , |z| < 1. (4) n=0 (q; q)n n=0 [n]q ! Copyright © by the paper’s authors. Copying permitted for private and academic purposes. In: L. Ferrari, M. Vamvakari (eds.): Proceedings of the GASCom 2018 Workshop, Athens, Greece, 18–20 June 2018, published at http://ceur-ws.org 172 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   n n x q(2) θx Y B fX (x) = (1 + θq j−1 )−1 , x = 0, 1, . . . , n, (5) x q j=1 and   n+x n+x−1 x q(2) θx Y NB fX (x) = (1 + θq j−1 )−1 , x = 0, 1, . . . , (6) x q j=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   n−x BS n Y fX (x) = θx (1 − θq j−1 ), x = 0, 1, . . . , n, (7) x q j=1 and   n N BS n+x−1 x Y fX (x) = θ (1 − θq j−1 ), x = 0, 1, . . . , (8) x q j=1 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. 2 Preliminaries Let v be a probability measure in R with finite moments of all orders Z sm = xm dv(x). R Then there exist a sequence of normalized orthogonal polynomials {pm (x)} with respect to the measure v satisfying the recurrence relation xpm (x) = pm+1 (x) + am pm (x) + bm pm−1 (x), m ≥ 1, (9) 173 with initial conditions p0 (x) = 1, p1 (x) = x − a0 . Moreover, they satisfy the orthogonality relation Z pm (x)pν (x)dv(x) = δmν b1 b2 · · · bm , m, ν ≥ 0 (10) S where δmν the Kronecker delta. The polynomials {pm (x)} depend on the moment sequence {sm }m≥0 and they can be obtain from the formula s0 s1 . . . sm s1 s2 . . . sm+1 . . . . s b1 b2 · · · bm pm (x) = . . . . , (11) Dm−1 Dm . . . . sm−1 sm . . . s2m−1 1 x . . . xm where Dm = det({si+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 bm = 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 = a0 and σ 2 = b1 . If am = 0 then all moments of odd order are zero Z s2m+1 = x2m+1 dv(x) = 0. x∈R Also, from the recurrence relation (9) the following representation of orthogonal polynomials is derived pm (x) = 1/2 x + a1 b2 0 ... ... 0 1/2 1/2 .. b2 x + a2 b3 0 . 1/2 .. .. .. 0 b3 x + a3 . . . p0 (x) . . (12) .. .. .. .. .. . . . . 0 .. .. .. 1/2 . . . x + bm−1 cm 1/2 0 ... ... 0 bm x + bm (see Szegö( [Sze59], p.374). Note that the probability measure v is uniquely determined if the coefficinets am and bm in the rec- curence 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 am and bm given in the next table respectively. 174 Table 1 : q-Classical Orthogonal Polynomials: Little q-Jacobi, affine q-Krawtchouk and little q-Laguerre/Wall Little q-Jacobi pLitJ m (x; a, b; q) q m (1+a2 bq m+1 +a(1−(1+b)q m −(1+b)q m −(1+b)q m+2 +bq 2m+1 )) am (1−abq 2m )(1−abq 2m+2 ) m+1 m m m m m m bm −aq (1−q )(1−aq )(1−bq )(1−abq )(c−abq )(1−cq ) (1−abq 2m )2 (1−abq 2m−1 )(1−abq 2m+1 ) Affine q-Krawtchouk pAf f m (x; p, n,  q)  am 1 − (1 − q m−n )(1 − pq m+1 ) − pq m−n (1 − q m ) bm pq m−n (1 − q m )(1 − pq m )(1 − q m−n−1 ) Little q-Laguerre/Wall pLLW m (x, a; q) am q m (1 − aq m+1 ) + aq m (1 − q m ) bm 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.   n n + g(y) − 1 Y fYN BS (y) = θg(y) (1 − θq j−1 ), 0 < θ < 1, 0 < q < 1, g(y) q j=1 y = [0]q , [1]q , [2]q , . . . , (13) where ln(1 − (1 − q)y) g(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. fYN BS (y) is induced by the normalized linear transfor- mation of the little q-Jacobi orthogonal polynomials, say pJm (y; a, b, q), 0 < q < 1, given by (−1)m (abq m+1 ; q)m LitJ pJm (y; a, b, q) = m pm (y; a, b, q), (14) q ( 2 ) (1 − q)m (aq; q) m where y = [x]q , x = 0, 1, . . . , and pLitJ 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.   n−g(y) n Y fYBS (y) = θg(y) (1 − θq j−1 ), 0 < θ < 1, 0 < q < 1, g(y) q j=1 y = [0]q , [1]q , [2]q , . . . , (15) 175 where ln(1 − (1 − q)y) g(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. fYBS (y) is induced by the normalized linear transforma- tion of deformed affine q-Krawtchouk orthogonal polynomials, say pAK m (y; p, q, n), 0 < q < 1, given by pAK m (y; p, n, q) = (1 − q)−m (pq, q −n ; q)m pAf f −n m (q y; p, n, q), (16) f −n where y = [x]q , x = 0, 1, . . . , and pAf m (q 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. λg(y) fYE (y) = Eq (−λ) , 0 < q < 1, 0 < λ(1 − q) < 1, [g(y)]q ! y = [0]q , [1]q , [2]q , . . . , (17) where ln(1 − (1 − q)y) g(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. fYE (y) is induced by the normalized linear transformation of the little q-Laguerre/Wall orthogonal polynomials, say pLm (y; a, q), 0 < q < 1, given by m pL m (y; a, q) = (−1)m (aq; q)m (1 − q)−m q ( 2 ) pLLW m (y; a, q), (18) where y = [x]q , x = 0, 1, . . . , and pLLW 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 dis- tribution, 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 q- negative 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. 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 defined by Y Y WG (M ) = W (i, j) wi hi,jiM i6∈M 176 and the corresponding generating function in m variables including the vertex and edge weights is defined by X Y Y P (G; w1 , w2 , . . . , wm ) = (−1)|M | W (i, j) wi M hi,jiM i6∈M with |M | the number of edges in M , summing over all matchings M of G . Let Lm 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 wi , i = 1, 2, . . . , m. Note that wi and W (i, i + 1) are bounded sequences in i, i = 1, 2, . . . . Kyriakoussis and Vamvakari [KV05], setting Ln = P (Lm ; w1 , w2 , . . . , wm ) X Y Y = (−1)|M | W (i, j) wi (19) M hi,jiM i6∈M where |M | the number of edges in M , have proved the following proposition. Proposition 3.5. The generating function of matching sets in paths, Lm , satisfies the recurrence relation Lm+1 = wn+1 Lm − W (m, m + 1)Lm−1 , m = 0, 1, 2, . . . (20) with initial conditions L−1 = 0 and L0 = 1. Remark 3.6. 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