NEVANLINNA THEORY OF THE ASKEY-WILSON DIVIDED DIFFERENCE OPERATOR YIK-MAN CHIANG AND SHAO-JI FENG Abstract. This paper establishes a version of Nevanlinna theory based on Askey-Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C. A second main theorem that we have derived allows us to define an Askey-Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole- scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey-Wilosn type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey-Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey-Wilson. This paper concludes with an application to difference equations generalising the Askey-Wilson second-order divided difference equation. Contents 1. Introduction 2 2. Askey-Wilson operator and Nevanlinna characteristic 6 3. Askey-Wilson type Nevanlinna theory – Part I: Preliminaries 8 4. Logarithmic difference estimates and proofs of Theorem 3.2 and 3.1 10 5. Askey-Wilson type counting functions and proof of Theorem 3.3 22 6. ProofoftheSecondMaintheorem3.5 25 7. Askey-Wilson type Second Main theorem – Part II: Truncations 27 8. Askey-Wilson-Type Nevanlinna Defect Relation 29 9. Askey-Wilson type Nevanlinna deficient values 31 arXiv:1502.02238v4 [math.CV] 3 Feb 2018 10. The Askey-Wilson kernel and theta functions 33 11. Askey-Wilson type Five-value theorem 37 12. Applications to difference equations 39 13. Concluding remarks 44 Appendix A. Proof of theorem 2.1 45 References 47 1991 Mathematics Subject Classification. Primary 33D99, 39A70, 30D35; Secondary 39A13. Key words and phrases. Nevanlinna theory, Askey-Wilson operator, Deficiency, difference equations. This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (600806, 16306315). The second author was also partially supported by National Natural Science Foundation of China (Grant No. 11271352) and by the HKUST PDF Matching Fund. 1 2 YIK-MANCHIANGANDSHAO-JIFENG 1. Introduction Without loss of generality, we assume q to be a complex number with q < 1. Askey and Wilson evaluated a q beta integral ([6, Theorem 2.1]) that allows| | them to construct a family of orthogonal− polynomials ([6, Theorems 2.2–2.5]) which are eigen-solutions of a second order difference equation ([6, 5]) now bears their names. § The divided difference operator q that appears in the second-order difference equation is called Askey-Wilson operatorD . These polynomials, their orthogonality weight, the difference operator and related topics have found numerous applications and connections with a wide range of research areas beyond the basic hypergeo- metric series. These research areas include, for examples, Fourier analysis ([11]), interpolations ([39], [31]), combinatorics ([18]), Markov process ([12], [48]), quan- tum groups ([35], [43]), double affine Hecke (Cherednik) algebras ([16], [34]). In this paper, we show, building on the strengths of the work of Halburd and Korhonen [23], [24] and as well as our earlier work on logarithmic difference esti- mates ([19], [20]), that there is a very natural function theoretic interpretation of the Askey-Wilson operator (abbreviated as AW operator) q and related topics. It is not difficult to show that the AW operator− is well-definedD on meromorphic functions. In particular, we show that− there is a Picard theorem associates with the Askey-Wilson operator just as the classical Picard theorem is associated with the conventional differential operator f ′. Moreover, we have obtained a full-fledged Nevanlinna theory for slow-growing meromorphic functions with respect to the AW operator on C for which the associated Picard theorem follows as a special case,− just as the classical Picard theorem is a simple consequence of the classical Nevanlinna theory ([41], see also [42], [27] and [50]). This approach allows us to gain new insights into the q and that give a radically different viewpoint from the established views on theD value distribution properties of certain meromorphic functions, such as the Jacobi theta -functions, generating functions of certain or- thogonal polynomials that were used in L. J. Rogers’ derivation of the two famous Rogers-Ramanujan identities [44], etc. We also characterise the functions that lie in the kernel of the Askey-Wilson operator, which we can regard as the constants with respect to the AW operator. A value a which is not− assumed by a meromorphic function f is called a Picard (exceptional) value. The Picard theorem states that if a meromorphic f that has three Picard values, then f necessarily reduces to a constant. For each complex number a, Nevanlinna defines a deficiency 0 δ(a) 1. If δ(a) 1, then that means f rarely assumes a. In fact, if a is a Picard≤ value≤ of f, then∼ δ(a)=1. If f assumes a frequently, then δ(a) 0. Nevanlinna’s second fundamental theorem ∼ implies that a∈C δ(a) 2 for a non-constant meromorphic function. Thus, the Picard theorem follows easily.≤ For each a C, we formulate a q deformation of P ∈ − the Nevanlinna deficiency ΘAW(a) and Picard value which we call AW deficiency and AW Picard value respectively. Their definitions will be given in− 8. The − § AW-deficiency also satisfies the inequalities 0 ΘAW(a) 1. A very special but illustrative example for ≤a C to be≤ an AW Picard value of a certain f if the pre-image of a C assumes the∈ form, with some−z C, ∈ a ∈ 1 (1.1) x := z qn + q−n/z , n N 0 . n 2 a a ∈ ∪{ } This leads to ΘAW(a) = 1. NEVANLINNA THEORY & ASKEY-WILSON OPERATOR 3 We illustrate some such AW Picard values in the following examples from the viewpoint with our new interpretation.− Let us first introduce some notation. We define the q shifted factorials: − n (1.2) (a; q) := 1, (a; q) := (1 aqk−1), n =1, 2, , 0 n − ··· kY=1 and the multiple q shifted factorials: − k (1.3) (a , a , ,a ; q) := (a ; q) . 1 2 ··· k n j n j=1 Y Thus, the infinite product (a1, a2, ,ak; q)∞ = lim (a1, a2, ,ak; q)n ··· n→+∞ ··· always converge since q < 1. The infinite products| | that appear in the Jacobi triple-product formula ([21, p. 15]) ∞ 2 (1.4) f(x) = (q; q) (q1/2z, q1/2/z; q) = ( 1)kqk /2zk, ∞ ∞ − k=X−∞ 1 −1 can be considered as a function of x where x = 2 (z + z ). The corresponding (zero) sequence is given by 1 x := q1/2+n + q−1/2−n , n N 0 n 2 ∈ ∪{ } where z = (q1/2 + q−1/2)/2 (a =0) . Thus 0 is an AW Picard value of f when a − viewed as a function of x, and hence f has ΘAW(0) = 1. Our next example is a generating function for a class of orthogonal polynomials known as continuous q Hermite polynomials first derived by Rogers in 1895 [44] − ∞ 1 Hk(x q) k f(x)= iθ −iθ = | t , 0 < t < 1, (te , te ; q)∞ (q; q)k | | kX=0 where n (q; q)n i(n−2k)θ Hn(x q)= e , x = cos θ. | (q; q)k (q; q)n−k kX=0 The orthogonality of these polynomials were worked out by Askey and Ismail [5, 1983]. We easily verify that the is an AW Picard value of f when viewed as a functions of x with the pole-sequence∞ given by− 1 (1.5) x := t qn + q−n/t , n N 0 , n 2 ∈ ∪{ } −1 where za = (t + t )/2 (a = ). This implies ΘAW( ) = 1. Our third example has both∞ zeros and poles. It is∞ again a generating function for a more general class of orthogonal polynomials also derived by Rogers in 1895 [44]. That is, (βeiθt, βe−iθt; q) ∞ (1.6) H(x) := ∞ = C (x; β q) tn, x = cos θ, (eiθt, e−iθt; q) n | ∞ n=0 X 4 YIK-MANCHIANGANDSHAO-JIFENG where n (β; q)k(β; q)n−k Cn(x; β q)= cos(n 2k)θ | (q; q)k(q; q)n−k − k=0 Xn (β; q)k(β; q)n−k = Tn−2k(x) (q; q)k(q; q)n−k Xk=0 is called continuous q ultraspherical polynomials by Askey and Ismail [5]. Here the − Tn(x) denotes the n th Chebychev polynomial of the first kind. Rogers [44] used these polynomials to− derive the two celebrated Rogers-Ramanujan identities ∞ 2 ∞ 2 qn 1 qn +n 1 = and = . (q; q) (q; q5) (q4; q5) (q; q) (q2; q5) (q3; q5) n=0 n ∞ ∞ n=0 n ∞ ∞ X X One can find a thorough discussion about the derivation of these identities in An- drews [3, 2.5]. The zero-§ and pole-sequences of H(x) in the x plane are given, respectively, by − 1 (1.7) x := βt qn + q−n/(βt) , n N 0 . n 2 ∈ ∪{ } and (1.5). The point is that we have both 0 and to be the AW Picard values ∞ − according to our interpretation. Thus ΘAW(0) = 1 and ΘAW( ) = 1 for the generating function H(x). ∞ Our Askey-Wilson version of Nevanlinna’s second fundamental theorem (Theo- rem 7.1) for slow-growing meromorphic functions not belonging to the kernel of q also implies that D (1.8) ΘAW(a) 2. C ≤ aX∈ This new relation allows us to deduce a AW Picard theorem (Theorem 10.2): Suppose a slow-growing meromorphic function −f has three values a, b, c C such ∈ that ΘAW(a)=ΘAW(b)=ΘAW(c)=1.