Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.


Introduction
In 1941 J.L. Burchnall (1892Burchnall ( -1975 wrote a short paper [7] in which he developed a method to find the inverse to the 1938 Feldheim-Watson formula which we rederive in Section 3.1, and the Feldheim-Watson formula (1.1) is equivalent to the finite Chu-Vandermonde sum. In fact, Nielsen [23, p. 33, equations (5) and (6)] derived the This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html Feldheim-Watson and the Burchnall formulas already in 1918. It seems that Burchnall was not aware of Nielsen's paper. Nielsen used the recurrence relations, whereas Burchnall used an operational approach. In fact in his series of memoirs in the 1890s, L.J. Rogers calculated the linearization coefficients for the continuous q-ultraspherical polynomials and the limiting case of the continuous q-Hermite polynomials, see, e.g., [10,Section 8.5] and references given there. So (1.1) is a q → 1 limiting case of one of Rogers's results. Rogers used his results to prove the Rogers-Ramanujan identities, see, e.g., [10,Section 8.10].
Actually, Burchnall gives an operational formula for d dx −2x n in terms of the k-th derivatives.
Explicitly, see [7, equation (3)], 3) for f sufficiently differentiable. The important ingredient in Burchnall's derivation are the raising and lowering operators for the Hermite polynomials. In Section 2 we show that Burchnall's method extends to all polynomials in the Askey-scheme and its q-analogue, see, e.g., [18,19], using their raising and lowering operators. We generalize (1.3) in Theorem 2.1. We give many worked out cases of the analogue of Burchnall's identity up to the Wilson and the Askey-Wilson polynomials. We restrict ourselves to families of orthogonal polynomials with infinite support, although it is clear that the method works for finite discrete orthogonal polynomials as well, up to minor modifications. It should be noted that Carlitz [8, equation (4)] extends Burchnall's operational formula (1.3) to the case of Laguerre polynomials which we rederive in (3.5). The resulting analogue of (1.2) is (m + 1) n n! L which Carlitz [8] proves by induction on n. Gould and Hopper [12] give a joint extension of these operational formulas to the Gould-Hopper polynomials. Al-Salam [2] shows that the operational formulas of Carlitz and Gould-Hopper are equivalent. Moreover, Singh [25] has determined the extension to the Jacobi case, which we give in Section 3.3 for completeness. The extensions of Burchnall's results to the Laguerre and Jacobi case fit in the classical orthogonal polynomials as a part of the Askey scheme. This part is characterized by having the derivative as the lowering operator. The extension to Zernike polynomials is given in [1], where more references to the literature are given.
In this paper we make the operational formulation (1.3) and the analogue of the expansion formulas (1.2), (1.4) explicit for several classes in the Askey scheme and its q-analogue. For the classical orthogonal polynomials -Hermite, Laguerre, Jacobi -this is in Section 3, and we show that the integrated formulas in case of the Hermite and Laguerre polynomials are essentially given as a change of coordinates. As is clear from the above, several of these results for classical orthogonal polynomials can be traced back in the literature, see [1,2,7,8,12,23,25] and references given there. We show how to generalize these operational formulas and expansion identities to all of the families of orthogonal polynomials in the Askey scheme and its q-analogue. Replacing the derivative by the backward shift operator as the lowering operator, we get two versions of the operational identities and the expansion formulas for the corresponding classes of orthogonal polynomials: Meixner and Charlier polynomials, see Section 4. In these cases there are two inequivalent versions of the expansion formulas of type (1.2), (1.4). Using the divided difference operator δ δx we give the explicit formulas for the Meixner-Pollaczek polynomials in Section 5. For the Wilson polynomials related to the divided difference operator δ δx 2 the results are stated in Section 6. For the q-analogue we study the big q-Jacobi polynomials and the Askey-Wilson polynomials. For the big q-Jacobi polynomials we again get two versions of the operational identity and the expansion formula given in Section 8. By switching to the Askey-Wilson q-difference operator as a lowering operator, we may include the Askey-Wilson polynomials. This is in Section 9, with emphasis on the special case of the continuous q-Hermite polynomials.
Using the generalized Burchnall identity and the raising and lowering operators being each others adjoint on suitable weighted L 2 -spaces, we obtain an integrated version in Corollary 2.3. In various cases we can use the integrated version to find orthogonal polynomials with respect to a slightly modified weight, where the modification is governed by an eigenfunction to certain operators occurring in the generalized Leibniz rule. We use this to find the orthogonal polynomials with respect to the orthogonality measure modified by an exponential or a q-exponential function. Often it is possible to recognize the modified measure explicitly in typically the same class. In this way we obtain expressions for orthogonal polynomials in terms of orthogonal polynomials with different parameters. In particular, for the modification by e −xt we can understand the orthogonal polynomials for the measure e −xt dµ(x), where the measure µ corresponds to the orthogonality measure for a family of orthogonal polynomials in the Askey-scheme. By the work of Flaschka, Moser and others this is related to the Lax pair for the Toda lattice, see, e.g., [5,Section 4.6], [6,Section 2], [14,Section 2.8], [27,Section 2]. The recurrence coefficients for the monic orthogonal polynomials xp n (x; t) = p n+1 (x; t) + b n (t)p n (x; t) + c n (t)p n−1 (x; t) for the modified weight satisfy the Toda lattice equationṡ (1.5) This approach works for the Hermite, Laguerre, Krawtchouk, Meixner, Charlier, and Meixner-Pollaczek polynomials, and this is listed in Proposition 7.1. Proposition 7.1 is well-known, and Zhedanov [26] derives the result from the requirement that the Lax pair (L, M ) and the time-derivativeL close up to a 3-dimensional Lie algebra. In this context, Proposition 7.1 follows from Lie algebra representations, see, e.g., [5,Section 4.6] for the link of the Toda lattice to simple Lie algebras, and its relation to orthogonal polynomials, see, e.g., [20] for the link to the corresponding polynomials. For the q-exponential functions e q and E q we find expansions of the big q-Jacobi polynomials in terms of big q-Laguerre polynomials. We also find the inverse formulas in this way. However, there seems no integrable system associated to these expansions. This paper deals with the scalar case. In a companion paper [15] we also use Burchnall type identities for matrix-valued orthogonal polynomials in order to give a non-trival solution to the non-abelian Toda lattice analogue of (1.5) as introduced by Gekhtman [11].
The contents of the paper are related to the role of the analogue of differentiation in the Askey-scheme and its q-analogue. First, in Section 2 we write down the general set-up as motivated by Burchnall's paper [7] and extending it. In Section 3 we look at those polynomials that correspond to the (ordinary) derivative, which are the classical polynomials of Hermite, Laguerre and Jacobi. We recover many of the known results in this way. By specializing we also recover expansions for Laguerre and Hermite polynomials, which can be considered as special cases of more general convolution identities [20]. These expansions are related to explicit solutions of the Toda lattice (1.5). In Section 4 we consider the shift operator as the analogue of the derivative, and we consider the families of Meixner and Charlier polynomials. In Section 5 we consider the Meixner-Pollaczek polynomials, and in the following Section 6 we consider the Wilson polynomials. In Section 7 we summarize the explicit solutions of the Toda lattice arising in this way. In the remaining sections we focus on the q-Askey scheme. In Section 8 we consider the q-Hahn scheme with the big q-Jacobi polynomials on top. Here we give several expansions for polynomial families in the q-Hahn scheme. In Section 9 we discuss the top-level of the Askey-Wilson polynomials.

Generalized Burchnall-type identities
In this section we show how the Burchnall identity for the Hermite polynomials as well as Burchnall's approach [7] can be generalized to families of orthogonal polynomials for which raising and lowering operators exist. In particular we give an expansion for a class of functions in terms of the polynomials involved. Note that all the polynomial families in the Askey-scheme and its q-analogue with infinite support fulfill the assumptions in this section up to Corollary 2.3, so that in particular Theorem 2.1 and Corollaries 2.2, 2.3 are valid in all cases studied in this paper.
Let the general orthogonal polynomials satisfy Here µ (ν) is a positive Borel measure on the real line for which all moments exist. The parameter ν is contained in some (multivariable) parameter set V ⊂ R n . We write a combination of parameters additively for the Askey scheme, but it is more convenient to write it multiplicatively for the q-Askey scheme. We assume that the measure µ (ν) has infinite support, but similar results for finite discrete orthogonal polynomials can be obtained up to some modifications. Next we assume that for two elements ν, ν + σ ∈ V of the parameter space we have (densely defined) raising operators R ν : L 2 µ (ν+σ) → L 2 µ (ν) . Here σ is a fixed element in R n , so that ν ∈ V implies that ν + Nσ ⊂ V. In particular, we assume R ν maps polynomials of degree n to polynomials of degree n + 1. The (densely defined) adjoint L ν = (R ν ) * : L 2 µ (ν) → L 2 µ (ν+σ) is assumed to be a lowering operator, i.e., mapping polynomials of degree n to polynomials of degree n − 1. So in particular, for all f ∈ Dom(R ν ) ⊂ L 2 µ (ν+σ) and all g ∈ Dom(L ν ) ⊂ L 2 µ (ν) , and in particular we assume that the polynomials are contained in the domains of R ν and L ν . It follows that for ν ∈ V we can take where 1(x) = 1 is the constant function. Note that in general [R ν+kσ , R ν+lσ ] = 0, so the order in (2.2) is relevant. Indeed, iterating (2.1) we find for polynomials f , g that 3) Take f = 1, g(x) = x k , k < n, so that L ν+(n−1)σ L ν+(n−2)σ · · · L ν+σ L ν g = 0 as a polynomial of negative degree k − n. Taking n , x k L 2 (µ (ν+σ) ) = 0, hence L ν p  Next we assume that the raising operators have a specific form. So we assume that there exists a function w ν for ν ∈ V so that w ν > 0 for µ (ν) -a.e. and that and ∂ is an operator independent of ν. In particular, we assume that w ν is in the domain of ∂.
Typically, µ (ν) is absolutely continuous with respect to Lebesgue measure or a counting measure (independent of ν), and the Radon-Nikodym derivative is w (ν) . So Typically, ∂ is a lowering operator. Moreover, we assume of the existence of a Leibniz rule for the operator ∂ of the form where α n k are constants, like binomial coefficients, η is a fixed operator (e.g., in order to accommodate the Askey-Wilson q-difference operator), and η can also be the identity. Here f and g are such that all expressions are well-defined. In general, we assume that η is an invertible homomorphism, so η(f g) = η(f )η(g). The T k,n are analogues of differential operators, and for the general case we have to allow for n-dependence.
For a suitable function f , such as a holomorphic function on a sufficiently large domain, an entire function or a polynomial, we have and rewriting proves Theorem 2.1.
Theorem 2.1. For f a suitable function, such as, e.g., a polynomial or a holomorphic function, we have using the notation and assumptions as above. Take 2), then we find from Theorem 2.1 and (2.2) the following polynomial identity, again using the notation of this section.
We can next use Theorem 2.1 in (2.3) to get Here we assume that each of the integrals converges. In particular, assuming that L ν = D is a lowering operator independent of ν, we obtain Corollary 2.3.
Assuming the convergence of the integrals and with the notation and assumptions as above, we have Corollary 2.3 in case of the Hermite polynomials is not contained in Burchnall's paper [7]. Note that for g(x) = x p with p < n the right hand side of Corollary 2.3 is zero.

Example: classical orthogonal polynomials
For the classical orthogonal polynomials in the Askey scheme, see [18,19], i.e., for the Jacobi, Laguerre and Hermite polynomials, we see that in Section 2 all the assumptions are fulfilled. Moreover, we can take D = ∂ = d dx , so that Leibniz formula is the usual one with α n k = n k , η is the identity, and T k,n = d k dx k = D k is a power of the lowering operator.

Example: Hermite polynomials
In this case we discuss Burchnall's motivating example of the Hermite polynomials [7], and we show how to extend some of Burchnall's results using the generating function (3.2) for Hermite polynomials. In this case the parameter set V = {0} as in Section 2 is trivial. So σ = 0, and w(x) = exp −x 2 is independent of ν, and so is Then the polynomials of Section 2 identify with the standard Hermite polynomials up to a sign; (3)]. Corollary 2.2 then gives (1.2), which was derived by Burchnall [7, equation (5)] and Nielsen [23, p. 22]. For this derivation we use that d . For the Hermite polynomials, Burchnall's identity (1.3) only involves the Hermite polynomials. This gives the opportunity to elaborate a bit more on Burchnall's identity. Multiply (1.3) by t n /n! and sum over n ∈ N to obtain after interchanging summation and using the generating function for the Hermite polynomials, see, e.g., [ where Z m (A, B) is a homogeneous polynomial of degree m in the non-commuting variables A and B given in terms of commutators;

So for this choice of
which gives the same result as (3.1) when acting on f .
Note that Corollary 2.3 after multiplication by n! gives Multiplying by t n , summing over n and using the generating function (3.2) gives On the other hand, from the integrated version (3.3) we see that H n x − 1 2 t also form orthogonal polynomials with respect to e −xt e −x 2 . So they are equal, up to a constant which can be determined by considering leading coefficients. This gives
for ν > −1. Multiplying (3.6) by u n , dividing by (−1) n n!, summing over n and using the generating function see, e.g., [19, equation (1.11.10)], gives for sufficiently smooth functions f and g and −1 < u < 1. Note that (3.8) is the integrated version, and it can be proved directly. From (3.8), i.e., Corollary 2.3 for the Laguerre polynomials, we can obtain an expression for the orthogonal polynomials with respect to e −xt x ν e −x on [0, ∞) by taking f (x) = exp(−xt) in (3.6), and take the polynomial g(x) = x p , with p < n. This gives the right hand side of as the orthogonal polynomials with respect to e −xt x ν e −x on [0, ∞). On the other hand, by a straightforward calculation using (3.8), the polynomials on the left hand side are also orthogonal with respect to the same weight. Hence, (3.9) follows up to a constant. This constant is determined by evaluating at x = 0. The result (3.9) is a convolution type identity, and can be directly proved using the generating function (
The integrated version follows from Corollary 2.3: so that we find only partial orthogonality, since the sum over k gives a polynomial of degree 2n, instead of n. For a detailed study of the orthogonal polynomials with respect to the Toda modification of the Jacobi weight e −xt (1 − x) α (1 + x) β on [−1, 1], we refer to Basor, Chen and Ehrhardt [6], where the relation to the Painlevé V equation is discussed.

The backward shift operator ∇
In this section we consider the results of Section 2 in case the derivative ∂ is given by the backward shift operator ∇, defined by ∇f (x) = f (x) − f (x − 1). In the Askey scheme, this corresponds to the families of Meixner and Charlier polynomials. The Krawtchouk polynomials form a family of finite discrete orthogonal polynomials, also contained in the part of the Askey scheme corresponding to ∇. We leave the Krawtchouk case to the reader, but we include the case in Proposition 7.1 for completeness. In order to apply the results of Section 2 we need to have the Leibniz formula (2.4) explicitly: where Sf (x) = f (x − 1). This follows by induction on n, since S and ∇ commute. Note that, upon comparing (4.1) with (2.4), we see that there are two choices for the homomorphism η of Section 2; η = 1 or η = S. Different choices lead to different expansions.

The Meixner polynomials
In the notation of Section 2 we have V = {(β, c) ∈ R 2 | β > 0, 0 < c < 1}, and we let σ = (1, 0), then ν ∈ V implies ν + nσ ∈ V for all n ∈ N. Put so that the orthogonality measure is a discrete measure supported in N with weight at x ∈ N equal to w(x; β, c). Then the raising operator R ν corresponds to R β, The adjoint L ν corresponds to −∆ is independent of ν, where ∆f (x) = f (x + 1) − f (x) is the forward shift operator.

Corollary 2.2 gives, after a simplification,
Applying Corollary 2.3 with f (x) = e −xt , which is a joint eigenfunction of ∇, ∆ and S, and g(x) = x p , p < n, we see that the polynomials of degree n n k=0 n k are orthogonal with respect to the discrete measure h → ∞ x=0 w(x; β, c)e −xt h(x), for which all moments exist for t > ln(c). Since this measure is again the measure for the Meixner polynomials with parameters (β, ce −t ) we find where the constant is obtained by evaluating at x = 0, since M n (0; β, c) = 1. Note that (4.4) is a convolution identity, which can be obtained directly from the generating function, see, e.g., In particular, we find an explicit solution to the Toda lattice (1.5) by looking at the threeterm recurrence relation for the monic version of M n (c; β, ce −t ). This in particular gives the explicit solution for the Meixner case of Proposition 7.1.

The Leibniz rule with η = S
Comparing (4.1) with (2.4) we can also take ∂ = ∇, η = S, α n k = n k , and T k,n = ∇ k in the general set-up. Then we identify The operational form of Theorem 2.1 is which gives an alternative expansion for the same operator as in (4.2).
Rewrite T k,n = ∇ k = (−1) k S k ∆ k , so that by [19, Note that (4.3) and (4.6) give different expansions for the same Meixner polynomial. Using Corollary 2.3 with f (x) = e −xt we find that the orthogonal polynomials for the Toda modification have an expansion as in the right hand side. Since we know the corresponding orthogonal polynomials as Meixner poynomials with parameters (β, ce −t ) we get

The Charlier polynomials
The orthogonality measure for the Charlier polynomials is a discrete measure supported on N and with corresponding weights w a (x) = a x x! , x ∈ N. In the notation of Section 2 we have V = {a > 0} ⊂ R. In this case the raising operator is independent of a, so that σ = 0. So and L = D = −∆. Then the polynomials p (ν) n correspond precisely to the Charlier polynomials C n (x; a) = 2 F 0 −n, −x; −; −1 a as in [19,Section 1.12]. It should be noted that these results can also be obtained by a limit transition from the corresponding results for the Meixner polynomials.

The Leibniz rule with η = 1
Here we give the analogues as in Section 4.1.1. So take ∂ = ∇, η = 1, α n k = n k , and T k,n = S n−k ∇ k = (−1) k S n ∆ k . Then we identify wν (x) with 1. So the operational formula of Theorem 2.1 gives  Using Corollary 2.3 with f (x) = e −xt we find an explicit expansion for the polynomials orthogonal with respect to w a (x)e −xt for x ∈ N. Since these are the polynomials C n (x; ae −t ) we obtain, cf. Section 4.1.1, This is a convolution type identity, which can be obtained from the generating function for Charlier polynomials; ∞ n=0 z n n! C n (x; a) = e z 1 − z a x , see, e.g., [19, equation (1.12.11)], and the Taylor expansion for the exponential function. It also follows from (4.4) by taking the limit from Meixner to Charlier polynomials, see [19, equation (2.9.2)].
In particular, we find an explicit solution to the Toda lattice (1.5) by looking at the three-term recurrence relation for the monic version of C n (x; ae −t ). This in particular gives the explicit solution for the Charlier case of Proposition 7.1.

The Leibniz rule with η = S
As in Section 4.1.2 we can also take ∂ = ∇, η = S, α n k = n k , and T k,n = ∇ k in the general setup. Then we identify So the operational formula of Theorem 2.1 gives the alternative to (4.8).
up to a constant which can be determined by considering leading coefficients. Then (4.12) can be proved using the same generating function for the Charlier polynomials and the binomial theorem. Again, (4.10) and (4.12) give different exansions for the same polynomial.

The difference operator δ δx
The difference operator δ δx is defined by In order to apply the results of Section 2 we need to have the Leibniz formula (2.4) explicitly: where S ± f (x) = f x ± 1 2 i , assuming that the functions f and g are defined in a sufficiently large strip | z| < 1 2 n + ε for ε > 0. This follows by induction on n, since S ± and δ δx commute. In this case the Leibniz formula is symmetric in shifting x to x ± 1 2 i, so it suffices to consider only one of the two possibilities. We take η = S − , α n k = n k , ∂ = δ δx , T k,n = (S + ) n−k δ δx k .

Burchnall's identity for the Wilson polynomials
As stated in Section 2 the Burchnall results can be derived for any polynomial family in the Askey scheme and its q-analogue. In this section we present the result for the top level of the Askey-scheme.
In this case is the parameter space. The parameter space can be made more general, see, e.g., [19, Section 1.1], but the results can be directly extended to the more general parameter space by analytic extension in the parameter, and we stick to the real parameters. The shift corresponds to σ = 1 2 , 1 2 , 1 2 , 1 2 , see, e.g., [19, equation (1.1.7)]. The weight function is, up to a constant independent of a and a change of variable, ω(x; a) = ω(x; a, b, c, d) Then the Wilson polynomials W n (·; a) satisfy the orthogonality ∞ 0 W n x 2 ; a W m x 2 ; a 2ixω(x; a)dx = 0, n = m, So the raising operator R ν in Section 2 corresponds to By n of Section 2 correspond precisely to the Wilson polynomials.
By induction we see that the Leibniz formula for δ δx 2 is where S ± f (x) = f x ± 1 2 i , assuming that the functions f and g are defined in a sufficiently large strip | z| < 1 2 n + ε. We then identify α n k with n k , η with S − and T k,n = (S + ) n−k δ δx 2 k .
We check that Then Theorem 2.1 gives the operational expansion Corollary 2.2 gives the following expansion for the Wilson polynomials: It is also possible to consider the Leibniz formula (6.1) with η identified with S + , cf. Section 4. However, by the symmetry for the Wilson polynomials, the results are equivalent.

Relation with the Toda lattice
In Section 1 we recalled that the Toda equations (1.5) are related to the modification of the orthogonality measure for orthogonal polynomials by e −xt . Considering Corollary 2.3 we see that we can obtain the orthogonal polynomials for the modified measure e −xt dµ (ν) assuming that (a) is a polynomial of degree at most k, and (b) f (x) = e −xt is an eigenfunction of T k,n as arising in the Leibniz formula (2.4).
Let us first consider condition (b). The T k,n follow from the Leibniz formula (2.4) for ∂, the operator from the raising operator for the orthogonal polynomials. Then we check the Askeyscheme and its q-analogue to see for which e −xt is an eigenfunction for these operators, and we only consider measures with infinite support. In the Askey-scheme, this only occurs if ∂ is the derivative d dx , corresponding to the Hermite, Laguerre and Jacobi polynomials; the backward shift operator ∇, corresponding to the Charlier and Meixner polynomials; the difference operator δ δx , corresponding to the Meixner-Pollaczek and continuous Hahn polynomials. The condition (a) is satisfied for the Hermite, Laguerre, Charlier, Meixner and Meixner-Pollaczek polynomials, as follows from the observations in Sections 3, 4, 5. In all these cases we find an explicit expression for the polynomials with respect to e −xt dµ (ν) (x) using Corollary 2.3. However, in all of these cases the measure e −xt dµ (ν) (x) can be identified with a related measure in the same family from the Askey scheme, i.e., one can glue e −xt onto the measure and obtain a known orthogonality measure.
Proposition 7.1. The following functions solve to Toda lattice equations (1.5): Note that all solutions of Proposition 7.1 correspond to solutions of the Toda lattice equations (1.5) which are separated. These solutions were obtained by Kametaka [16,17].
Proof . The first five cases follow from the observations in Sections 3, 4, 5 and the corresponding recurrence coefficients in the relations for the monic orthogonal polynomials as in, e.g., [19].
The final one for the Krawtchouk polynomials follows by observing that the exponential modification of Krawtchouk weight N In the q-Askey-scheme none of these operators works to give explicit orthogonal polynomials for the modification of the weight by e −xt . In Section 8 we consider the case with a q-exponential, but this is not related to the Lax pair for the Toda lattice. 8 Burchnall's identities for the q-Hahn scheme Our next objective is to give the Burchnall type identities for orthogonal polynomials from the q-Askey-scheme. In this section we determine the corresponding identities in the q-Hahn scheme, as a subscheme of the q-Askey scheme, see [21,Chapter 3]. The big q-Jacobi polynomials are on top of the q-Hahn scheme, and the raising and lowering operators are given in terms of the q-derivative D q ; The Leibniz rule for the q-derivative is, see [10, Exercise 1.12(iv)], where (T q f )(x) = f (xq) is the q-shift and n k q = (q;q)n (q;q) k (q;q) n−k is the q-binomial. In the context of Section 2 we have ∂ = D q , and D = D q −1 .

Big q-Jacobi polynomials
The big q-Jacobi polynomials are defined by P n (x; a, b, c; q) = 3 ϕ 2 q −n , abq n+1 , x aq, cq ; q; q (8.2) and using the q-integral f cq k q k , the big q-Jacobi polynomials are orthogonal with respect to the weight function w(x; a, b, c; q) = (x/a, x/c; q) ∞ (x, xb/c; q) ∞ with respect to the discrete measure f → aq cq f (x)w(x; a, b, cd; q)d q x, which corresponds to the measure dµ (ν) of Section 2. In the correspondence of Section 2, let ν correspond to (a, b, c), and ν + kσ corresponds to q k a, q k b, q k c . Let V = {(a, b, c) | 0 < a < q −1 , 0 < b < q −1 , c < 0}, so that the condition on V as in Section 2 is satisfied. Now R ν corresponds to R a,b,c = M −1 a,b,c • D q • M aq,bq,cq : n (x) of Section 2 correspond to the big q-Jacobi polynomials (aq, cq; q) n (ac) n q n(n+1) (1 − q) n P n (x; a, b, c; q) by [19, equation (3.5.9-10)]. The adjoint L ν of R ν equals D q −1 , since D q −1 is the formal adjoint of ∂ = D q for the (unweighted) q-integral. Put η = T q , α n k = n k q , T k,n = D k q , so that corresponds to w xq k ; q k a, q k b, q k c; q w(x; a, b, c; q) = (x, xb/c; q) k .
Using (8.3), Theorem 2.1 gives the operational formula R a,b,c R aq,bq,cq · · · R aq n−1 ,bq n−1 ,cq n−1 f (x) = n k=0 n k q (x, xb/c; q) k aq k+1 , cq k+1 ; q n−k T q P n−1 (·; aq, bq, cq; q) =⇒ D k q P n (·; a, b, c; q) = (−1) k q 1 2 k(k+1) q −n , abq n+1 ; q k (1 − q) n (aq, cq; q) k T k q P n−k ·; aq k , bq k , cq k ; q (8.5) taking into account the relation D q T q = qT q D q . After simplification, Corollary 2.2 is P n+m (x; a, b, c; q) = n∧m k=0 q −n , q −m , abq 2n+m+1 , x, xb/c; q k q, aq, cq, aq n+1 , cq n+1 ; q k (ac) k q k 2 +2k+nk × P n−k xq k ; aq k , bq k , cq k ; q P m−k xq k ; aq n+k , bq n+k , cq n+k ; q . (8.6) Note that the left hand side is obviously symmetric in n and m, but the right hand side is not. Corollary 2.3 gives aq cq n k=0 n k q (x, xb/c; q) k aq k+1 , cq k+1 ; q n−k for p < n. Take f (x) = e q (−tx) = 1 (−tx;q)∞ in (8.7) assuming t ∈ −a −1 q −N ∪ −c −1 q −N , so that there are no poles in the support of the measures involved. Then f is an eigenfunction of the q-derivative; D k q f = −t k f , so that aq cq n k=0 n k q (x, xb/c; q) k aq k+1 , cq k+1 ; q n−k acq 2k n−k q (n−k)(n−k+1) (1 − q) n−k × P n−k xq k ; aq k , bq k , cq k ; q −t (1 − q) k 1 (−tx; q) ∞ x p w(x; a, b, c; q)d q x = 0 for p < n. Take the limit b ↓ 0, i.e., specialize to the big q-Laguerre polynomials [19,Section 3.11]. So in particular, we find an expression for the orthogonal polynomials with respect to the measure aq cq 1 (−tx; q) ∞ w(x; a, 0, c; q)d q x = aq cq w(x; a, −tc, c; q)d q x, which is again the orthogonality measure for a big q-Jacobi polynomial. Hence, we find the following equality up to a constant, which is determined by considering leading coefficients or by evaluating at x = 1, after replacing t by −b/c: n k=0 (q −n , x; q) k (q, aq, cq; q) k (−abq n ) k q 1 2 k 2 + 3 2 k P n−k xq k ; aq k , 0, cq k ; q = P n (x; a, b, c; q). (8.8) The result (8.8) seems not be easily provable using a generating function, see, e.g., (3.9) for the classical counterpart provable as a convolution identities from generating functions. The other natural candidate for f in (8.7) is f (x) = E q (−xt) = (xt; q) ∞ . Note that so that we can only get orthogonal polynomials for the measure modified by the big q-exponential E q (−xt) if cancellation in (x,xb/c;q) k (tx;q) k occurs. This happens precisely for t = 1 or t = b/c. In particular, taking t = b/c we see that the polynomials of degree n defined by n k=0 n k q (x; q) k aq k+1 , cq k+1 ; q n−k acq 2k n−k q (n−k)(n−k+1) (1 − q) n−k P n−k xq k ; aq k , bq k , cq k ; q −b/c 1 − q Since this is the measure for the big q-Laguerre polynomials, i.e., big q-Jacobi polynomials (8.2) with b = 0, see, e.g., [19,Section 4.11], we also know the orthogonal polynomials as big q-Laguerre polynomials. This gives the following expansion up to a constant, which is determined by evaluating at x = 1: n k=0 (q −n , x; q) k (q, aq, cq; q) k (−ab) k q k(k+n+1) P n−k xq k ; aq k , bq k , cq k ; q = P n (x; a, 0, c; q) (8.9) and (8.9) can be considered as a kind of inverse to (8.8).