On the properties of special functions on the linear-type lattices

arXiv:1107.5898v1 [math.CA] 29 Jul 2011 ON THE PROPERTIES OF SPECIAL FUNCTIONS ON THE LINEAR-TYPE LATTICES R. ´ ALVAREZ-NODARSE AND J. L. CARDOSO Abstract. We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to q-classical polynomials are given. 2000 Mathematics Subject Classification 33D15, 33D45 1. Introduction The study of the so-called q-special functions has known an increasing interest in the last years due its connection with several problems in mathematics and mathematicalphysics (see e.g. [3, 6, 8, 13, 17]). A systematic study starting from the second order linear difference equation that such functions satisfy was started by Nikiforov and Uvarov in 1983 and further developed by Atakishiyev and Suslov (for a very nice reviews see e.g. [7, 13, 16]). Of particular interest is the so-called q-classical polynomials (see e.g. [5]) introduced by Hahn in 1949 which are polynomials on the lattice qs. Our main aim in this paper is to present a constructive approach for generating recurrence relations and ladder-type operators for the difference analogues of special functions of hypergeometric type on the linear-type lattices. Here we will focus our attention on functions defined on the q-linear lattice (for the linear lattice x(s) = ssee [4] and references therein, and for the continuous case see e.g. [18]). Therefore we will complete the work started in [16] where few recurrence relations where obtained. In fact we will prove, by using the q-analoge of the technique introduced in [4] for the discrete case (uniform lattice), that the solutions (not only the polynomial ones) of the difference equation on the q-linear lattice x(s) = c1qs+c2satisfy a very general recurrent-difference relation from where several well known relations (such as the three-term recurrence relation and the ladder-type relations) follow. The structure of the paper is as follows: In section 2 the needed results and notations from the q-special function theory are introduced. In sections 3 and 4 the general theorems for obtaining recurrences relations are presented. In section 5 the special case of classical q-polynomials are considered in details and some examples are worked out in details. 2. Some preliminar results Here we collect the basic background [1, 13, 16] on q-hypergeometric functions needed in the rest of the work. The hypergeometric functions on the non-uniform lattice x(s) are the solutions of the second order linear difference equation of hypergeometric type on non-uniform lattices σ(s)∆ ∆x(s−1 2)∇y(s) ∇x(s)+τ(s)∆y(s) ∆x(s)+λy(s) = 0, σ(s) = eσ(x(s)) −1 2eτ(x(s))∆xs−1 2, τ(s) = eτ(x(s)), (1) Key words and phrases. q-hypergeometric functions, difference equations, recurrence relations, qpolynomials. MAY 15, 2011 1 2 R. ´ ALVAREZ-NODARSE AND J. L. CARDOSO where ∆y(s) := y(s+ 1) −y(s), ∇y(s) := y(s)−y(s−1), are the forward and backward difference operators, respectively; eσ(x(s)) and eτ(x(s)) are polynomials in x(s) of degree at most 2 and 1, respectively, and λis a constant. Here we will deal with the linear and q-linear lattices, i.e., lattices of the form x(s) = c1s+c2or x(s) = c1(q)qs+c2(q),(2) respectively, with c16= 0 and c1(q)6= 0 . We will define the k-order difference derivative of a solution y(s) of (1) by y(k)(s) := ∆(k)[y(s)] = ∆ ∆xk−1(s) ∆ ∆xk−2(s)... ∆ ∆x(s)[y(s)], where xν(s) = x(s+ν 2). It is known [13] that y(k)(s) also satisfy a difference equation of the same type. Moreover, for the solutions of the difference equation (1) the following theorem holds Theorem 2.1. [12, 16] The difference equation (1) has a particular solution of the form yν(z) = Cν ρ(z) b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z)](ν+1) ,(3) if the condition σ(s)ρν(s)∇xν+1(s) [xν−1(s)−xν−1(z+ 1)](ν+1)  b a = 0, is satisfied, and of the form yν(z) = Cν ρ(z)ZC ρν(s)∇xν+1(s) [xν(s)−xν(z)](ν+1) ds, (4) if the condition ZC ∆s σ(s)ρν(s)∇xν+1(s) [xν−1(s)−xν−1(z+ 1)](ν+1) = 0,(5) is satisfied. Here Cis a contour in the complex plane, Cνis a constant, ρ(s)and ρν(s)are the solution of the Pearson-type equations ρ(s+ 1) ρ(s)=σ(s) + τ(s)∆x(s−1 2) σ(s+ 1) =φ(s) σ(s+ 1), ρν(s+ 1) ρν(s)=σ(s) + τν(s)∆xν(s−1 2) σ(s+ 1) =φν(s) σ(s+ 1), (6) where τν(s) = σ(s+ν)−σ(s) + τ(s+ν)∆x(s+ν−1 2) ∆xν−1(s),(7) νis the root of the equation λν+ [ν]qαq(ν−1)eτ′+ [ν−1]qeσ′′ 2= 0,(8) and [ν]qand αq(ν)are the q-numbers [ν]q=qν/2−q−ν/2 q1/2−q−1/2, αq(ν) = qν/2+q−ν/2 2,∀ν∈C,(9) respectively. The generalized powers [xk(s)−xk(z)](ν)are defined by [xk(s)−xk(z)](ν)= (q−1)νcν 1qν(k−ν+1)/2qνz Γq(s−z+ν) Γq(s−z), ν ∈R,(10) for the q-linear (exponential) lattice x(s) = c1qs+c2and [xk(s)−xk(z)](ν)=cν 1 Γ(s−z+µ) Γ(s−z), ν ∈R, SPECIAL FUNCTIONS ON THE LINEAR-TYPE LATTICES 3 for the linear lattice x(s) = c1s+c2, respectively. For the definitions of the Gamma and the q-Gamma functions see, for instance, [6]. Remark 2.2. For the special case when ν∈N, the generalized powers become [xk(s)−xk(z)](n)=(−1)ncn 1q−n(n−1)/2qn(z+k/2)(qs−z;q)n, [xk(s)−xk(z)](n)=cn 1(s−z)n, for q-linear and linear lattices, respectively. We will need the following straightforward proposition which proof we omit here (see e.g. [1, 16]) Proposition 2.3. Let µand νbe complex numbers and mand kbe positive integers with m≥k . For the q-linear lattice x(s) = c1qs+c2we have (1) [xµ(s)−xµ(z)](m) [xν(s)−xν(z)](m)=qm(µ−ν) 2, (2) [xµ(s)−xµ(z)](m) [xµ(s)−xµ(z)](k)= [xµ(s)−xµ(z−k)](m−k), (3) [xµ(s)−xµ(z)](m) [xν(s)−xν(z)](k)=qk(µ−ν) 2[xµ(s)−xµ(z−k)](m−k), (4) [xµ(s)−xµ(z)](m+1) [xµ−1(s+ 1) −xµ−1(z)](m)=xµ−m(s)−xµ−m(z), (5) [xµ(s)−xµ(z)](m+1) [xµ−1(s)−xµ−1(z)](m)=xµ−m(s+m)−xµ−m(z). To obtain the result for the linear lattice one only has to put in the above formulas q= 1. 3. The general recurrence relation in the linear-type lattices In this section we will obtain several recurrence relations for the solutions (3) and (4) of the difference equation (1) in the linear-type lattices (2). Since the equation (1) is linear we can restrict ourselves to the canonical cases x(s) = qsand x(s) = s. Let us define the functions1 Φν,µ(z) = b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z)](µ+1) (11) and Φν,µ(z) = ZC ρν(s)∇xν+1(s) [xν(s)−xν(z)](ν+1) ds. (12) Notice that the functions yνand the functions Φν,µ are related by the formula yν(z) = Cν ρ(z)Φν,ν(z).(13) Lemma 3.1. For the functions Φν,µ(z)the following relation holds ∇zΦν,µ(z) = [µ+ 1]q∇xν−µ(z)Φν,µ+1(z),(14) where [t]qdenotes the symmetric q-numbers (9). 1Obviously the functions (3) correspond to the functions (11), whereas the functions yνgiven by (4) correspond to those of (12). 4 R. ´ ALVAREZ-NODARSE AND J. L. CARDOSO Proof. We will prove it for the functions (11). The other case is analogous. Using (10), one gets ∇zΦν,µ(z) = b−1 X s=a∇z ρν(s)∇xν+1(s) [xν(s)−xν(z)](µ+1) ! = b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z)](µ+1) −ρν(s)∇xν+1(s) [xν(s)−xν(z−1)](µ+1) ! = b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z−1)](µ)1 xν(s)−xν(z)−1 xν(s)−xν(z−1−µ) = b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z−1)](µ) xν(z)−xν(z−1−µ) (xν(s)−xν(z))(xν(s)−xν(z−1−µ)) = b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z)](µ+2) (xν(z)−xν(z−1−µ)) Since x(s)−x(s−t) = [t]q∇xs−t−1 2we then have ∇zΦν,µ(z) = b−1 X s=a ρν(s)∇xν+1(s) [xν(s)−xν(z)](µ+2) [µ+ 1]q∇xνz−µ 2 =[µ+ 1]q∇xν−µ(z)Φν,µ+1(z) which is (14).  From (14) follows that ∆zΦν,µ(z) = [µ+ 1]q∆xν−µ(z)Φν,µ+1(z+ 1). Next we prove the following lemma that is the discrete analog of the Lemma in [14, page 14]. Lemma 3.2. . Let x(z)be x(z) = qzor x(z) = z. Then, any three functions Φνi,µi(z), i= 1,2,3, are connected by a linear relation 3 X i=1 Ai(z)Φνi,µi(z) = 0,(15) with non-zero at the same time polynomial coefficients on x(z),Ai(z), provided that the differences νi−νjand µi−µj,i, j = 1,2,3, are integers and that the following condition holds2 xk(s)σ(s)ρν0(s) [xν0−1(s)−xν0−1(z)](µ0) s=b s=a = 0, k = 0,1,2,..., (16) when the functions Φνi,µiare given by (11) and ZC ∆s xk(s)σ(s)ρν0(s)ds [xν0−1(s)−xν0−1(z)](µ0)= 0, k = 0,1,2,..., (17) when Φνi,µiare given by (12). Here ν0is the νi,i= 1,2,3, with the smallest real part and µ0is the µi,i= 1,2,3, with the largest real part. Proof. Since in [4] we have proved the case when x(s) = s(the uniform lattice) we will restrict here to the case of the q-linear lattice x(s) = c1qs+c2). Moreover, we will give the proof for the case of functions of the form (11), the other case is completely equivalent. Using the identity ∇xνi+1(s) = qνi−ν0 2∇xν0+1(s), 2In some cases this condition is equivalent to the condition x(s)kσ(s)ρν0(s)|s=b s=a= 0, k= 0,1,2,.... SPECIAL FUNCTIONS ON THE LINEAR-TYPE LATTICES 5 as well as (3) of Proposition 2.3, we have 3 X i=1 Ai(z)Φνi,µi(z) = 3 X i=1 Ai(z) b−1 X s=a ρνi(s)∇xνi+1(s) [xνi(s)−xνi(z)](µi+1) = b−1 X s=a 3 X i=1 Ai(z)ρνi(s)∇xνi+1(s) [xνi(s)−xνi(z)](µi+1) = b−1 X s=a 1 [xν0(s)−xν0(z)](µ0+1) × 3 X i=1 Ai(z)q(µi+1)(ν0−νi) 2[xν0(s)−xν0(z−µi−1)](µ0−µi)ρνi(s)∇xνi +1(s)! = b−1 X s=a ρν0(s)∇xν0+1(s) [xν0(s)−xν0(z)](µ0+1) × 3 X i=1 Ai(z)qµi(ν0−νi) 2[xν0(s)−xν0(z−µi−1)](µ0−µi)ρνi(s) ρν0(s)!. Using the Pearson-type equation (6) we obtain ρνi(s) = φ(s+ν0)φ(s+ν0+ 1) . . . φ(s+νi−1)ρν0(s),(18) so 3 X i=1 Ai(z)Φνi,µi(z) = b−1 X s=a ρν0(s)∇xν0+1(s) [xν0(s)−xν0(z)](µ0+1) Π(s) where Π(s) = 3 X i=1 Ai(z)qµi(ν0−νi) 2[xν0(s)−xν0(z−µi−1)](µ0−µi)× φ(s+ν0)φ(s+ν0+ 1) ···φ(s+νi−1) . (19) Let us show that there exists a polynomial Q(s) in x(s) (in general, Q≡Q(z, s) is a function of zand s) such that ρν0(s)∇xν0+1(s) [xν0(s)−xν0(z)](µ0+1) Π(s) =∆ "ρν0(s−1) [xν0−1(s)−xν0−1(z)](µ0)Q(s)# =∆ "σ(s)ρν0(s) [xν0−1(s)−xν0−1(z)](µ0)Q(s)#. (20) If such polynomial exists, then, taking the sum in sfrom s=ato b−1 and using the boundary conditions (16) we obtain (15). To prove the existence of the polynomial Q(s) in the variable x(s) in (20) we write σ(s+ 1)ρν0(s+ 1) [xν0−1(s+ 1) −xν0−1(z)](µ0)Q(s+ 1) −σ(s)ρν0(s) [xν0−1(s)−xν0−1(z)](µ0)Q(s) = ρν0(s) [xν0(s)−xν0(z)](µ0+1) σ(s+ 1)ρν0(s+1) ρν0(s) [xν0(s)−xν0(z)](µ0+1) [xν0−1(s+1)−xν0−1(z)](µ0)Q(s+ 1)− σ(s)[xν0(s)−xν0(z)](µ0+1) [xν0−1(s)−xν0−1(z)](µ0)Q(s). From (4) and (5) of Proposition 2.3, and using (6), the above expression becomes ρν0(s) [xν0(s)−xν0(z)](µ0+1) {φν0(s) [xν0−µ0(s)−xν0−µ0(z)] Q(s+ 1)− σ(s) [xν0−µ0(s+µ0)−xν0−µ0(z)] Q(s)}. 6 R. ´ ALVAREZ-NODARSE AND J. L. CARDOSO Thus (σ(s) + τν0(s)∇xν0+1(s)) [xν0−µ0(s)−xν0−µ0(z)] Q(s+ 1)− σ(s) [xν0−µ0(s+µ0)−xν0−µ0(z)] Q(s) = ∇xν0+1(s)Π(s). (21) Since ∇xν0+1(s) is a polynomial of degree one in x(s) , xk(s) and τν0(s) are polynomials of degree at most one in x(s), and σ(s) is a polynomial of degree at most two in x(s), we conclude that the degree of Q(s) is, at least, two less than the degree of Π(s), i.e., deg Q≥deg Π −2. Moreover, equating the coefficients of the powers of x(s) = qson the two sides of the above equation (21), we find a system of linear equations in the coefficients of Q(s) and the coefficients Ai(z) which have at least one unknown more then the number of equations. Notice that the coefficients of the unknowns are polynomials in qz, so that after one coefficient is selected the remaining coefficients are rational functions of qz, therefore after multiplying by the common denominator of the Ai(z) we obtain the linear relation with polynomial coefficients on x≡x(z) = qz. This completes the proof.  The above Lemma when q→1 and x(s) = sleads to the corresponding result on the uniform lattice x(s) [4]. 3.1. Some representative examples. In the following examples, and for the sake of simplicity, we will use the notation σ(s) = aq2s+bqs+c, τ(s) = dqs+e, φν(s) = σ(s)+τν−1(s)∇xν(s) = fq2s+gqs+h. (22) Example 3.3. The following relation holds A1(z)Φν,ν−1(z) + A2(z)Φν,ν +A3(z)Φν+1,ν(z) = 0, where the coefficients A1,A2and A3, are polynomials in x≡x(z) = qz,given by A1(z) = −eq ν 2+ b+eq1 2−q−1 2 a+dq1 2−q−1 2dq ν 2+a[ν]q+dqν+a[2ν]qqν 2+z, A2(z) = c(dqν+a[2ν]q) a+dq1 2−q−1 2+ b+eq1 2−q−1 2 q1 2−q−1 2 qν+a qνa+dq1 2−q−1 2 qz+dqν+a[2ν]qq2z, A3(z) = −dq ν 2+a[ν]q a+dq1 2−q−1 2, where a,b,c,d, and e, are the coefficients of σand τ(22). Proof. Using the notations of Lemma 3.2 we have ν1=ν,ν2=ν,ν3=ν+1 , µ1=ν−1 , µ2=νand µ3=ν, thus ν0=νand µ0=ν . By (19) Π(s) = A1qs+ν 2−qz− ν 2+A2+A3q− ν 2ha+dq1 2−q−1 2q2ν+2s+b+eq1 2−q−1 2qν+s+ci. (23) On the other hand, from (21) and because Q(s) = kis a constant –notice that deg (Π) = 2– we have ∇xν0+1(s)Π(s) = knha+dq1 2−q−1 2q2ν+2s+b+eq1 2−q−1 2qν+s+ci(qs−qz)− aq2s+bqs+c(qν+s−qz)(24) where kis an arbitrary constant. Introducing (23) in (24), using the identity ∇xν0+1(s) = qν 2q1 2−q−1 2qs and comparing the coefficients of the powers of x(s) = qswe get a linear system of three equations with four variables A1,A2,A3and k. Choosing k= 1 and solving the corresponding system we get, after some simplifications, the coefficients A1,A2and A3. SPECIAL FUNCTIONS ON THE LINEAR-TYPE LATTICES 7 In the next examples, since the technique is similar to the previous one we will omit the details. Example 3.4. The following relation holds A1(z)Φν,ν(z) + A2(z)Φν,ν+1(z) + A3(z)Φν+1,ν+1(z) = 0 , where the coefficients A1,A2and A3, are polynomials in x≡x(z) = qz,given by A1(z) = fa−f q2νqz+a g q −f b qν+1 , A2(z) = q−ν 2−1a−f q2νf q2z+g qz+1 +h q2, A3(z) = √q(a q −f qν), where a,b,c,f,gand h, are the coefficients of σand φν(22). Example 3.5. The following relation holds A1(z)Φν−1,ν−1(z) + A2(z)Φν,ν−1(z) + A3(z)Φν,ν (z) = 0 , where the coefficients A1,A2and A3, are polynomials in x≡x(z) = qz,given by A1(z) =q−1 2 −νnfq2zh−a2hq4+agbqν+4 −q2ν+2 ag2q−2fah +fb2 −fgbq3ν+1 q2−q−1+fq4νg2(q−1) q−fhi+ gqz+1h−a2hq5+aqν+2 gbq3+fhq2−fh−q2ν+2fah +fgb +ag2q2− f2ah−b2+gbq−fah+fq3νq2g2q−fh+gb−g2+fh+f2hq4νq2−q−1i −a2h2q6+aghqν+5 (bq +gq −g) + fghq3ν+4 (gq +b−g)−f2h2q4ν+2 −hq2ν+3ag2q3+fgbq2+fg2q2−2fahq +fb2q−2fg2q−fgb +fg2o, A2(z) = q− ν 2−qν 2fq2z+gqz+1 +hq2fqzfq2ν−aq2+fqν+1(gq+b−g)−agq3, A3(z) =f(fqν−aq)hfq2z+hq2fq2ν−aq2+gqz+1fqνqν+q−1−aq3i, where a,b,c,f,gand h, are the coefficients of σand φν(22). Example 3.6. The following relation holds A1(z)Φν−1,ν−1(z) + A2(z)Φν,ν(z) + A3(z)Φν,ν+1(z) = 0 , where the coefficients A1,A2and A3, are polynomials in x≡x(z) = qz,given by A1(z) = a2hq4−agbqν+3+q2ν+2fb2−2fah+ag2−fgbq3ν+1+f2hq4ν, A2(z) = q−1 2fqν−aq2fqz+2ν−aqz+2 +gq2ν+1 −bqν+2, A3(z) = −qv−3 2f q2v−aq2qv+1 −1gqz+1 +fq2z+hq2, where a,b,c,f,gand h, are the coefficients of σand φν(22). Example 3.7. The relation A1(z)Φν,ν−1(z) + A2(z)Φν,ν(z) + A3(z)Φν+1,ν+1(z) = 0 , is verified when the polynomial coefficients A1,A2and A3, in the variable x≡x(z) = qz, are given by A1(z) = qν+1 2fqz+νfq2ν−gqν+b−a−f(h−b)q2ν+1 −aq (gqν−h), A2(z) = q−z+ν+1 2qzfq2ν−a+qν(gqν−b)fq2z+gqz+1 +hq2, A3(z) = q2z(fqν−aq) + qz+νq(gqν−aq −b)−fqνqν+1 −q−1+ qν+1(h−b)qν+1 +gqν−h, where a,b,c,f,gand h, are the coefficients of σand φν(22). 8 R. ´ ALVAREZ-NODARSE AND J. L. CARDOSO 4. Recurrences involving the solutions yν In [16] the following relevant relation was established ∆(k)yν(s) = C(k) ν ρk(s)Φν , ν−k(s),(25) where C(k) ν=Cν k−1 Y m=0 αq(ν+m−1)eτ′+ [ν+m−1]qeσ′′ 2. This relation is valid for solutions of the form (3) and (4) of the difference equation (1). In the following, y(k) n(s) denotes the k-th differences ∆(k)yn(s). Theorem 4.1. In the same conditions as in Lemma 3.2, any three functions y(ki) νi(s), i= 1,2,3, are connected by a linear relation 3 X i=1 Bi(s)y(ki) νi(s) = 0,(26) where the Bi(s),i= 1,2,3, are polynomials. Proof. From Lemma 3.2 we know that there exists three polynomials Ai(s), i= 1,2,3 such that 3 X i=1 Ai(s)Φνi,νi−ki(s) = 0, then, using the relation (25), we find 3 X i=1 Ai(s)(C(k) ν)−1ρki(s)y(ki) νi(s) = 0. Now, dividing the last expression by ρk0(s), where k0= min{k1, k2, k3}, and using (18) we obtain 3 X i=1 Bi(s)y(ki) νi(s) = 0, Bi(s) = Ai(s)(C(k) ν)−1φ(s+k0)···φ(s+ki−1), which completes the proof.  Corollary 4.2. In the same conditions as in Lemma 3.2, the following three-term recurrence relation holds A1(s)yν(s) + A2(s)yν+1(s) + A3(s)yν−1(s) = 0, with polynomial coefficients Ai(s),i= 1,2,3. Proof. It is sufficient to put k1=k2=k3= 0, ν1=ν,ν2=ν+ 1 and ν3=ν−1 in (26).  Corollary 4.3. In the same conditions as in Lemma 3.2, the following ∆-ladder-type relation holds B1(s)yν(s) + B2(s)∆yν(s) ∆x(s)+B3(s)yν+m(s) = 0, m ∈Z,(27) with polynomial coefficients Bi(s),i= 1,2,3. Proof. It is sufficient to put k1=k3= 0, k2= 1, ν1=ν2=νand ν3=ν+min (26).  SPECIAL FUNCTIONS ON THE LINEAR-TYPE LATTICES 9 Notice that for the case m=±1 (27) becomes B1(s)yν(s) + B2(s)∆yν(s) ∆x(s)+B3(s)yν+1(s) = 0,(28) e B1(s)yν(s) + e B2(s)∆yν(s) ∆x(s)+e B3(s)yν−1(s) = 0,(29) with polynomial coefficients Bi(s) and e Bi(s), i= 1,2,3. The above relations are usually called raising and lowering operators, respectively, for the functions yν. Let us now obtain a raising and lowering operators for the functions yνbut associated to the ∇/∇x(s) operators. We start applying the operator ∇/∇x(s) to (13) ∇ ∇x(s)yν(s) = ∇ ∇x(s)Cν ρ(s)Φν,ν(s) =1 ∇x(s)CνΦνν(s)1 ρ(s)−1 ρ(s−1)+Cν ρ(s−1)∇Φνν(s), or, equivalently, ∇Φνν ∇x(s)=ρ(s−1) Cν ∇yν(s) ∇x(s)−Φνν(s) ∇x(s)ρ(s−1) ρ(s)−1. By Lemma (3.2) with ν1=µ1=ν2=ν , µ2=ν+ 1 and ν3=µ3=ν+m, there exist polynomial coefficients on x(s), Ai(s), i= 1,2,3, such that A1(s)Φν,ν(s) + A2(s)Φν,ν+1(s) + A3(s)Φν+m,ν+m(s) = 0. From (14) Φν,ν+1(s) = 1 [ν+ 1]q ∇Φν,ν ∇x(z)=1 [ν+ 1]q ∇Φν,ν ∇x(z). Therefore A1(s)Φν,ν +A2(s) [ν+ 1]qρ(s−1) Cν ∇yν ∇x(s)−Φνν(s) ∇x(s)ρ(s−1) ρ(s)−1 +A3Φν+m,ν+m= 0. Using now the Pearson equation (6) and dividing by ρ(s) we get A1(s)yν(s) + A2(q) [ν+ 1]qσ(s) φ(s−1) ∇yν ∇x(s)−yν(s) ∇x(s)σ(s) φ(s−1) −1 +A3 Cν Cν+m yν+m(s) = 0 . Multiplying both sides by [ν+ 1]qφ(s−1) , A1(s)[ν+ 1]qφ(s−1)yν(s) + A2(s)σ(s)∇yν ∇x(s)− A2(s)σ(s)−φ(s−1) ∇x(s)yν(s) + [µ+ 1]qCνC−1 ν+mA3φ(s−1)yν+m(s) = 0. Thus we have proven the following Theorem 4.4. In the same conditions as in Lemma 3.2, the following ∇-ladder-type relation holds C1(s)yν(s) + C2(s)∇yν(s) ∇x(s)+C3(s)yν+m(s) = 0, m ∈Z,(30) with polynomial coefficients Ci(s),i= 1,2,3.