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1 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P. -56 Advances in Mathematics Contents lists available at ScienceDirect Advances in Mathematics Nevanlinna theory of the Askey Wilson divided difference operator 0 0 Yik-Man Chiang a,,, Shaoji Feng b, 4 a Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 4 5 b Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 5 6 Beijing, 00080, PR China 6 8 a r t i c l e i n f o a b s t r a c t 8 0 Article history: This paper establishes a version of Nevanlinna theory based Received 8 September 06 on Askey Wilson divided difference operator for meromorphic 0 functions of finite logarithmic order in the complex plane C. Available online xxxx A second main theorem that we have derived allows us to Communicated by George Andrews define an Askey Wilson type Nevanlinna deficiency which MSC: gives a new interpretation that one should regard many 4 important infinite products arising from the study of basic 4 30D35 Accepted 3 January D05 hypergeometric series as zero/pole-scarce. That is, their 39D45 zeros/poles are indeed deficient in the sense of difference A3 Nevanlinna theory. A natural consequence is a version 6 7 of Askey Wilson type Picard theorem. We also give an 7 Keywords: alternative and self-contained characterisation of the kernel 8 Nevanlinna theory 8 functions of the Askey Wilson operator. In addition we have 9 Askey Wilson operator established a version of unicity theorem in the sense of Askey 9 Deficiency 30 Wilson. This paper concludes with an application to difference Difference equations * Corresponding author addresses: machiang@ust.hk Y.-M. Chiang, fsj@amss.ac.cn S. Feng. Partially supported by Research Grants Council of the Hong Kong Special Administrative Region, China , , Partially supported by National Natural Science Foundation of China Grant No. 735 and by the HKUST PDF Matching Fund, / 08 Published by Elsevier Inc. 4 divided difference operator, Adv. Math. 08,

2 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P. -56 Y.-M. Chiang, S. Feng / Advances in Mathematics equations generalising the Askey Wilson second-order divided difference equation. 08 Published by Elsevier Inc.. Introduction Without loss of generality, we assume q to be a complex number with q <. Askey and Wilson evaluated a q-beta integral [6, Theorem.] that allows them to construct a family of orthogonal polynomials [6, Theorems..5] which are eigen-solutions of a second order difference equation [6, 5] now bears their names. The divided difference operator D q that appears in the second-order difference equation is called Askey Wilson operator. 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 hypergeometric series. These research areas include, for examples, Fourier analysis [], interpolations [38], [3], combinatorics [0], Markov process [], [44], quantum groups [34], [4], double affine Hecke Cherednik algebras [5], [33]. In this paper, we show, building on the strengths of the work of Halburd and Korhonen [3], [4] and as well as our earlier work on logarithmic difference estimates [7], [8], that there is a very natural function theoretic interpretation of the Askey Wilson operator abbreviated as AW-operator D q and related topics. It is not difficult to show that the AW-operator is well-defined 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 [40], see also [4], [7] and [46]. This approach allows us to gain new insights into the D q and that give a radically different viewpoint from the established views on the value distribution properties of certain meromorphic functions, such as the Jacobi theta-functions, generating functions of certain orthogonal polynomials that were used in L. J. Rogers derivation of the two famous Rogers Ramanujan identities [43], 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 δa. If δa, then that means f rarely assumes a. In fact, if a is a Picard value of f, then δa. If f assumes a frequently, then δa 0. Nevanlinna s second fundamental theorem implies that a C δa for divided difference operator, Adv. Math. 08,

3 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.3-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 3 a non-constant meromorphic function. Thus, the Picard theorem follows easily. For each a C, we formulate a q-deformation of the Nevanlinna deficiency Θ AW 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 Θ AW a. 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 a C, x n : za q n + q n /z a, n N {0}. 0 0 This leads to Θ AW a. 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: a; q 0 :, a; q n : 7 k 7 and the multiple q-shifted factorials: j Thus, the infinite product a, a,, a k ; q n : n aq k, n,,, a, a,, a k ; q lim a, a,, a k ; q n n + k a j ; q n. 3 always converge since q <. The infinite products that appear in the Jacobi triple-product formula [, p. 5] fx q; q q / z, q / /z; q 3 k 3 33 k q k / z k, 4 can be considered as a function of x where x z + z. The corresponding zero sequence is given by x n : q /+n + q / n, n N {0} where z a q / + q / / a 0. Thus 0 is an AW-Picard value of f when viewed as a function of x, and hence f has Θ AW 0. Our next example is a generating function for a class of orthogonal polynomials known as continuous q-hermite polynomials first derived by Rogers in 895 [43] divided difference operator, Adv. Math. 08,

4 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics k0 where H k x q fx te iθ, te iθ t k, 0 < t <, ; q q; q k H n x q n q; q n e in kθ, x cos θ. q; q k q; q n k 7 k0 7 The orthogonality of these polynomials were worded out by Askey and Ismail [5, 983]. 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 0 0 x n : t q n + q n /t, n N {0}, 5 where z a t + t / a. This implies Θ AW. 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 895 [43]. That is, Hx : βeiθ t, βe iθ t; q e iθ t, e iθ C n x; β q t n, x cos θ, 6 t; q n0 where C n x; β q n β; q k β; q n k cosn kθ q; q k q; q n k 6 k0 6 n β; q k β; q n k T n k x q; q k q; q n k 9 k0 9 is called continuous q-ultraspherical polynomials by Askey and Ismail [5]. Here the T n x denotes the n-th Chebychev polynomial of the first kind. Rogers [43] used these polynomials to derive the two celebrated Rogers Ramanujan identities q n q n +n 35 q; q n q; q 5 q 4 ; q 5 and q; q n q ; q 5 q 3 ; q n0 n0 36 One can find a thorough discussion about the derivation of these identities in Andrews [3,.5]. The zero- and pole-sequences of Hx in the x-plane are given, respectively, by x n : βt q n + q n /βt, n N {0}, 7 divided difference operator, Adv. Math. 08,

5 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.5-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 5 and 5. The point is that we have both 0 and to be the AW-Picard values according to our interpretation. Thus Θ AW 0 and Θ AW for the generating function Hx. Our Askey Wilson version of Nevanlinna s second fundamental theorem Theorem 7. for slow-growing meromorphic functions not belonging to the kernel of D q also implies that Θ AW a. 8 9 a C This new relation allows us to deduce a AW-Picard theorem Theorem 0.: Suppose a slow-growing meromorphic function f has three values a, b, c C such that Θ AW a Θ AW b Θ AW c. Then f lies in the kernel of D q. Note that, what Nevanlinna proved can be viewed when a meromorphic function has three Picard values then the function lies in the kernel of differential operator. By the celebrated Jacobi triple-product formula [4, p. 497], we can write the Jacobi theta-function ϑ 4 z, q + k n q k cos kz in the infinite product form ϑ 4 z, q q ; q q e iz, q e iz, q implying that it too has Θ AW 0 when viewed as a function fx of x. Since the fx is entire, so that the relationship 8 becomes Θ AW 0 a C Θ AW a. We deduce from this inequality that there could not be a non-zero a such that the theta function have fx n a only on a sequence {x n } of the form. Otherwise, it would follow from Theorem 8.4 that the theta function ϑ 4 would belong to the kernel ker D q, contradicting the kernel functions representation that we shall discuss in the next paragraph. The same applies to the remaining three Jacobi theta-functions. Intuitively speaking, the more zeros the function has out of the maximal allowable number of zeros of the meromorphic function can have implies the larger the AW-Nevanlinna deficiency Θ AW 0. That is, the function assumes x 0 less often in the AW-sense, even though the function actually assumes x 0 more often in the conventional sense. Thus, since the theta function assumes zero maximally, so it misses x 0 also maximally in the AW-sense. The following examples that we shall study in details in 9 show how zeros are missed/assumed in proportion to the maximally allowed number of zeros against their AW-deficiencies. That is, for a given integer n, n 39 f n x q k e iθ, q k e iθ ; q n+, Θ AW 0 n n, 40 n 40 k0 and divided difference operator, Adv. Math. 08,

6 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics n x q k e iθ, q k e iθ ; q n n, Θ AW 0 n. f k0 In [30, p. 365] Ismail has given an example of meromorphic function that belongs to ker D q : fx cos θ cos φ qe iθ+φ, qe iθ+φ ; q qe iθ φ, qe iθ φ ; q q / e iθ+φ, q / e iθ+φ ; q q / e iθ φ, q / e iθ φ ; q, 9 for a fixed φ. Let f belong to the kernel of D q, that is D q f 0. Then one can readily deduce from 8 that the f, when viewed upon as a function of θ, is doubly periodic, and hence must be an elliptic function in θ. 3 However, the authors could not find an explicit discussion of this observation in the literature. Here we offer an alternative and self-contained derivation to characterise these kernel functions when viewed as a function of x cos θ. Our Theorem 0. shows that all functions in the ker D q are essentially a product of functions of this form. Intuitively speaking, the functions that lie in the kernel must have zero- and pole-sequences described by. We utilise the linear structure of the ker D q to deduce any given number of linear combination of certain q-infinite products can again be expressed in terms of a single q-infinite product of the same form whose zero- and pole-sequences can again be described by see Theorems 0.3 and 0.4. Many important Jacobi theta-function identities such as the following well-known see [45] 0 0 and ϑ 4z ϑ 4 + ϑ z ϑ ϑ 3z ϑ 3 0 ϑ 3 z + y ϑ 3 z y ϑ ϑ 3y ϑ 3z + ϑ y ϑ z are of the forms described amongst our Theorems 0.3 and 0.4. The key to establishing a q-deformation of the classical Nevanlinna second fundamental theorem is based on our AW-logarithmic difference estimate of the proximity function for the meromorphic function f: 33 m r, D qfx O log r σ log +ε fx holds for all x r sufficiently large, where σ log is the logarithmic order of f Theorem 3.. This estimate is the key of our argument to establishing our AW-Nevanlinna theory see also [4], [3], [4]. We have also obtained a corresponding pointwise estimate Theorem 3. outside a set of finite logarithmic measure. 4 3 The authors are grateful for the referee who pointed out this fact. 4 divided difference operator, Adv. Math. 08,

7 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.7-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 7 Similar estimates for the simple difference operator fx fx+η fx for a fixed η 0, were obtained by Chiang and Feng in [7], [8], and by Halburd and Korhonen [3] independently in a slightly different form for finite-order meromorphic functions. Halburd et al. showed the same for the case of q-difference operator q fx fqx fx in [7] for zero-order meromorphic functions, whilst Cheng and Chiang [4] showed that a similar logarithmic difference estimate again holds for the Wilson operator. There has been a surge of activities in extending the classical Nevanlinna theory which is based on differential operator to various difference operators in recent years such as the ones mentioned above [3]. The idea has been extended to tropical functions [6], [36]. The original intention was to apply Nevanlinna theory to study integrability of non-linear difference equations [], [4]. But as it turns out that difference type Nevanlinna theories have revealed previously unnoticed complex analytic structures of seemingly unrelated subjects far from the original intention, such as the topic discussed in this paper. This paper is organised as follows. We will introduce basic notation of Nevanlinna theory and Askey Wilson theory in. The AW-type Nevanlinna second main theorems will be stated in 3 and 7. The definition of AW-type Nevanlinna counting function will also be defined in 7. The proofs of the logarithmic difference estimate and the truncated form of the second main theorem are given in 4 and 7 respectively. The AW-type Nevanlinna defect relations as well as an AW-type Picard theorem are given in 8. This is followed by examples constructed with arbitrary rational AW-Nevanlinna deficient values in 9. We characterize the transcendental functions that belongs to the kernel of the AW-operator in 0. These are the so-called AW-constants. We also illustrate how these functions are related to certain classical identities of Jacobi theta-functions there. It is known that the Askey Wilson orthogonal polynomials are eigen-functions to a second-order linear self-adjoint difference equation given in [6]. In we demonstrate that if two finite logarithmic order meromorphic functions such that the pre-images at five distinct points in C are identical except for an infinite sequences of the form as given in, then the two functions must be identical, thus giving an AW-Nevanlinna version of the well-known unicity theorem. We study the Nevanlinna growth of entire solutions to a more general second-order difference equation in than the Askey Wilson selfadjoint Strum Liouville type equation 7 using the tools that we have developed in this paper Askey Wilson operator and Nevanlinna characteristic Let fx be a meromorphic function on C. Let r x, then we denote log + r max{log r, 0}. We define the Nevanlinna characteristic of f to be the real-valued function where Tr, f : mr, f + Nr, f, 3 divided difference operator, Adv. Math. 08,

8 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics π r mr, f log + fre iθ dθ, Nr, f x ± 9 33 nt, f n0, f dt 4 t and nr, t denote the number of poles in { x < r}. The real-valued functions mr, f and Nr, f are called the proximity and integrated counting functions respectively. The characteristic function Tr, f is an increasing convex function of log r, which plays the role of log Mr, f for an entire function. The first fundamental theorem states that for any complex number c C Tr, c : T r, Tr, f + O 5 f c as r +. We refer the reader to Nevanlinna s [4] and Hayman s classics [7] see also [46] for the details of the Nevanlinna theory. We now consider the Askey Wilson operator. We shall follow the original notation introduced by Askey and Wilson in [6] see also alternative notation in [30, p. 300] with slight modifications. Let fx be a meromorphic function on C. Let x cos θ. We define fz f z + /z/ fx fcos θ, z e iθ. 6 That is, we regard the function fx as a function fz of e iθ z. Then for x ± the q-divided difference operator Dq f x : δ f q fq e iθ : fq e iθ δ q x ĕq e iθ ĕq e iθ, eiθ z, 7 where ex x is the identity map, is called the Askey Wilson divided difference operator. In these exceptional cases, we have D q f ± lim x ± Dq f x f ±q + q / instead. It can also be written in the equivalent form Dq f x : fq e iθ fq e iθ, x z + /z/ cos θ. 8 q q z /z/ Since there are two branches of z that corresponds to each fixed x, we choose a branch of z x + x such that x x as x and x / [, ]. We define the values of z on [, ] by the limiting process that x approaches the interval [, ] from above the real axis. Thus, z assumes the value z x + i x where x is now real and x. So we can guarantee that for each x in C there corresponds a unique z in C and z as x. Finally we note that if we know that fx is analytic at x, then lim Dq f x f x. q divided difference operator, Adv. Math. 08,

9 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.9-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 9 We can now define a polynomials basis 3 n 3 φ n cos θ; a : ae iθ, ae iθ ; q n 4 k r 5 0 k axq k + a q k ; 9 which plays the role of the x n in conventional differential operator. Askey and Wilson [6] computed that D q φ n x; a a qn φ n x; aq, 0 q for each integer n. Ismail and Stanton [3] established that if fx is an entire function satisfying log Mr, f lim sup log r c < log q, where Mr, f : max x r fx denotes the maximum modulus of f, then one has fx f k,φ ae iθ, ae iθ q k ; q k, f k, φ a k q kk /4 Dq k fx k q; q k where the f k, φ is the k-th Taylor coefficients and the x k is defined by x k : aq k/ + q k/ /a /, k 0. 3 We note, however, that the interpolation points x k are those points with k being even. We record here some simple observations about the operator D q acting on meromorphic functions, the justification of them will be given in the Appendix A. We first need the averaging operator [30, p. 30]: A q fx [ ] fq z + fq z. Theorem.. Let f be an entire function. Then A q f and D q f are entire. Moreover, if fx is meromorphic, then so are A q f and D q f. The useful product/quotient rule [30, p. 30] is given by D q f/g A q fd q /g + A q /gd q f. 4 Since we consider meromorphic functions in this paper so we extend the growth restriction on f from to those of finite logarithmic order [6], [9] defined by log Tr, f lim sup σ log f σ log < +. 5 r log log r divided difference operator, Adv. Math. 08,

10 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics It follows from an elementary consideration that the logarithmic order for transcendental function must have σ log f, and σ log f for rational functions and σ log 0 for constant functions. We note that the growth assumption in is a special case of our 5. Now let us suppose that fx is a meromorphic function that satisfies 5. Then f has order zero in the proper Nevanlinna order sense see [7]. It follows from a result by Miles [39] that f can be represented as a quotient f g/h where both g and h are entire functions and that each of them again satisfies 5. Thus the Askey Wilson operator is well-defined on the class of slow-growing finite logarithmic order meromorphic functions. We refer the reader to [8], [6], [9] and [8] for further properties of slow-growing meromorphic functions Askey Wilson type Nevanlinna theory part I: preliminaries Nevanlinna s second main theorem is a deep generalisation of the Picard theorem. Nevanlinna s second main theorem implies that for any meromorphic function f satisfies the defect relation c Ĉ δc. That is, if f a, b, c on Ĉ, then δa δb δc. This is a contradiction to Nevanlinna s defect relation. The proof of the Second Main Theorem is based on the logarithmic derivative estimates mr, f /f otr, f which is valid for all x r if f has finite order and outside an exceptional set of finite linear measure in general. We have obtained earlier that for a fixed η 0 and any finite order meromorphic function f of finite order σ, and arbitrary ε > 0 the estimate m fx + η/fx Or σ +ε [7] valid for all x r. Halburd and Korhonen [3] proved a comparable estimate independently for their pioneering work on a difference version of Nevanlinna theory [4] and their work on the integrability of discrete Painlevé equations [5]. Here we also have a AW-logarithmic difference lemma: Theorem 3.. Let fx be a meromorphic function of finite logarithmic order σ log 5 such that D q f 0. Then we have, for each ε > 0, that 33 holds for all x r > 0 sufficiently large. m r, D qfx O log r σ log +ε 6 fx This estimate is crucial to the establishment of the Nevanlinna theory in the sense of Askey Wilson put forward in this paper. In addition to the average estimate above, we have obtained a corresponding pointwise estimate of the logarithmic difference which holds outside some exceptional set of x : Theorem 3.. Let fx be a meromorphic function of finite logarithmic order σ log 5 such that D q f 0. Then we have, for each ε > 0, that divided difference operator, Adv. Math. 08,

11 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P. -56 Y.-M. Chiang, S. Feng / Advances in Mathematics log + D q fx O log r σ log +ε 7 fx holds for all x r > 0 outside an exceptional set of finite logarithmic measure. This estimate when written in the form D qfx exp [ log x σ log +ε ] fx should be compared to our earlier estimate fx + /fx exp [ x σ +ε] [7, Theorem 8.] and the classical estimate f x/fx x σ +ε of Gundersen [, Corollary ] for meromorphic functions of finite order σ, both hold outside exceptional sets of x of finite logarithmic measures. An analogue has been obtained recently by Cheng and the first author of this paper in [4] for the Wilson divided difference operator. However, unlike in all these estimates where Cartan s lemma was used in their derivations, our argument is direct and avoids the Cartan lemma. 0 0 We will prove the Theorem 3. and Theorem 3. in section 4. Theorem 3.3. Let f be a meromorphic function of finite logarithmic order σ log. Then, for each ε > 0, Nr, D q f Nr, f + O log r σ log +ε + Olog r. 8 We shall prove this theorem in 5. Theorem 3.4. Let f be a meromorphic function of finite logarithmic order σ log. Then, for each ε > 0, In particular, this implies Tr, D q f Tr, f + O log r σ log +ε + Olog r. 9 σ log D q f σ log f σ log. 30 Proof. We deduce from Theorem 3. and Theorem 3.3 that Tr, D q f m r, D qfx + mr, f + Nr, D q f fx 33 as required. mr, f + Nr, f + O log r σ log +ε + Olog r Tr, f + O log r σ log +ε + Olog r, We are now ready to state our first version of the Second Main Theorem whose proof will be given in 6. 3 divided difference operator, Adv. Math. 08,

12 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P. -56 Y.-M. Chiang, S. Feng / Advances in Mathematics Theorem 3.5. Suppose that fz is a meromorphic function of finite logarithmic order σ log 5 such that D q f 0 and let A, A,, A p p, be mutually distinct elements in C. Then we have for every ε > 0 mr, f + p mr, A ν Tr, f N AW r, f + O log r σ log +ε 3 6 ν 6 holds for all r x > 0, where N AW r, f : Nr, f Nr, D q f + N r, Logarithmic difference estimates and proofs of Theorem 3. and 3. We recall the following elementary estimate. 9 9 α log x x C α x + x x x c n <R. 33 D q f Lemma 4. [7]. Let α, 0 < α be given. Then there exists a constant C α > 0 depending only on α, such that for any two complex numbers x and x, we have the inequality In particular, C. x x 38 c n <R c n <R 40 α Lemma 4.. Let fx be a meromorphic function of finite logarithmic order σ log 5 such that D q f 0 and α is an arbitrary real number such that 0 < α <. Then there exist a positive constant C α such that for q / + q / x < R, we have log + D q fx 4 R q/ + q / x m R, f + m R, fx R x [R q / + q / x ] f + q / + q / x R x + R q / + q / x nr, f + nr, f + C α q / α + q / α x α + C α q / α x α + C α q / α x α x c n α x + cqq / z q / c n α + log, x cqq / z q / c n α where the {c n } denotes the combined zeros and poles sequences of f. 35 divided difference operator, Adv. Math. 08,

13 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.3-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 3 Proof. We start by expressing all the logarithmic difference in terms of complex variables x as well as in z in the Askey Wilson divided difference operator. So it follows from 8 that D q fx fq e iθ fq e iθ fx fxq q z /z/, 36 f[ q / z + q / z / ] f [ q / z + q / z / ] fxq q z /z/ q q z /z/ [ f q / z + q / z / ] 0 0 fx f[ q / z + q / z / ] fx where x z + /z/ cos θ and we recall that we have fixed our branch of z for the corresponding x in the above expressions. Let cq q / q / /. 38 We deduce from 36 that, by letting x and hence z to be sufficiently large log + D q fx log + fx q q z /z/ + log + f [ q / z + q / z / ] fx + log + f [ q / z + q / z / ] + log fx log + / cqz + log + f [ q / z + q / z / ] fx + log + f [ q / z + q / z / ] + log fx log f [ q / z + q / z / ] fx + log f [ q / z + q / z / ] fx + log. For x and hence z to be sufficiently large, 39 divided difference operator, Adv. Math. 08,

14 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics q ±/ z + q / z / q ±/ x + q / q ±/ z / q ±/ x < R. 40 It is obvious that x < R. We apply the Poisson Jensen formula see e.g., [7, p. ] to estimate the individual terms on the right-hand side of the above expression 39. Thus, f [ q / z + q / z / ] f log fx log [ q / z + q / z / ] log fx 8 π 8 Re log fre iφ iφ + q / z + q / z / R dφ π Re iφ q / z + q / z / π Re log fre iφ iφ + x R π Re iφ dφ x log R b µ q / z + q / z / R[q / z + q / z / b µ ] 6 b µ <R 6 log R ā ν q / z + q / z / R[q / z + q / z / a ν ] 8 a ν <R 8 log R b µ x + Rx b µ log R ā ν x. Rx a ν b µ <R a ν <R That is, f [ q / z + q / z / ] log fx log fre iφ Re iφ q / z + q / z x R dφ π Re iφ x[re iφ q / z + q / z /] 7 π log R b µ q / z + q / z / R b µ x 3 b µ <R 3 33 log Rq/ z + q / z / b µ Rx b µ 34 b µ <R 34 log R ā ν q / z + q / z / R ā ν x 37 a ν <R 37 + log Rq/ z + q / z / a ν Rx a ν a ν <R π 4 Re iφ [q / x + q / q / z ] 4 π log fre iφ R Re iφ x[re iφ q / x + q / q / z /] dφ 4 0 divided difference operator, Adv. Math. 08,

15 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.5-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 5 + log R b µ [q / x + q / q / / z ] R b µ x b µ <R log [q/ x + q / q / / z ] b µ x b µ 5 b µ <R 5 log R ā ν [q / x + q / q / / z ] R ā ν x 7 a ν <R 7 + log [q/ x + q / q / / z ] a ν x a ν 0 a ν <R 0 log fre iφ Re iφ [q / x + cq z ] R dφ π Re iφ x[re iφ q / x + cq z ] π log R b µ [q / x + cq z ] log [q/ x + cq z ] b µ x b µ R b b µ <R µ x b µ <R log R ā ν [q / x + cq z ] R + ā ν x 8 a ν <R a ν <R 8 8 π π log [q/ x + cq z ] a ν x a ν where we have made the substitution 38. We let x and hence z be sufficiently large, so we may assume that cqz < min q / x, q / x, q / x, q / x in the following calculations. We notice that the integrated logarithmic average term from 4 has the following upper bound log fre iφ Re iφ [q / x + cq z ] R dφ π Re iφ x[re iφ q / x + cq z ] Hence 4 becomes log fre iφ 4 R q / x π R x R q / x dφ 4 R q / x m R, f + m R,. R x R q / x f [ log f q / z + q / z / ] fx 4 R q / x m R, f + m R, 43 R x R q / x f 4 divided difference operator, Adv. Math. 08,

16 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics + log R b µ [q / x + cq z ] R b µ x + log [q/ x + cq z ] b µ x b µ log R ā ν [q / x + cq z ] R ā ν x + log [q / x + cq z ] a ν x a ν b µ <R b µ <R + 5 a ν <R a ν <R 5 Applying the Lemma 4. with α, to each individual term in the first summand of 43 with b ν < R yields log R b µ [q / x + cqz ] R b µ x 0 0 µ [ q b / x cqz ] µ [ q R b + b / x cqz ] µ [q / x + cqz ] R b µ x R q/ x R R q / x + R q/ x R R x q / x R x + R q / x Similarly, we have, for the third summand that for a µ < R, log R ā ν [q / x + cqz ] R ā ν x q / x R x + R q / x Again applying the Lemma 4. with 0 α < to each individual term in the second summand of 4 yields log q / x + cqz b µ x b µ q / x + cqz α q / x + cqz α C α + q / x + cqz b µ x b µ C α q / α x α x b µ α + q / x + cqz b µ α 33 Similarly, we have, for the fourth summand, log q / x + cqz a ν x a ν C α q / α x α x a ν α +. q / x + cqz a ν α divided difference operator, Adv. Math. 08,

17 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.7-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 7 Combining the inequalities 43, yields [ log f q / z + q / z / ] fx 4 R q / x m R, f + m R, R x [R q / x ] f + q / x R x + nr, f + nr, R q / x f + C α q / α x α x a ν α + q / x + cqz a ν α 0 0 a ν <R + C α q / α x α x b µ α + q / x + cqz b µ α 4 b µ <R 4 4 R q / x m R, f + m R, R x [R q / x ] f + q / x R x + nr, f + nr, R q / x f + C α q / α x α x c n α + q / x + cqz c n α c n <R where we have re-labelled all the zeros {a ν } and poles {b µ } by the single sequence {c n }. Replacing q by q in the 48, we obtain for x sufficiently large [ log f q / z + q / z / ] fx 4 R q / x m R, f + m R, R x [R q / x ] f + q / x R x + nr, f + nr, R q / x f + C α q / α x α x c n α + q / x cqz c n α c n <R 34 Substituting the 48 and 49 into 39 yields 35. Proof of Theorem 3.. In order to give an upper bound estimate for the last three summands of the above sum 35, we need to avoid exceptional sets arising from the sequence given by {d n } : {c n } {c n q / } {c n q / } divided difference operator, Adv. Math. 08,

18 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics Given ε > 0, let { [ d n E n r : r d n log σlog+ε d n + 3, d d n ] } n + log σlog+ε 50 d n + 3 and E n E n. Henceforth we consider the x / E. It is not difficult to see the inequality 0 0 holds for all x sufficiently large. Thus 7 c n <R 7 Similarly, we have x d n x dn x log σlog+ε x + 3, 5 x c n α α log α σlog+ε x + 3 x α nr, f + nr,. f 5 and x + cqq / z q / c n x q / c n cqq / z x q / c n cqq / z x log σlog+ε x + 3 cqq / z x 3 log σlog+ε x + 3, holds for x sufficiently large. Hence x cqq / z q / x c n 3 log σlog+ε x + 3 x + cqq / z q / c n 3α log ασlog+ε x + 3 α x α nr, f + nr, f 33 c n <R and c n <R x cqq / z q / c n 3α log ασlog+ε x + 3 α x α nr, f+nr,. f 56 We obtain from 35, after substituting 5, 55 56, the inequality divided difference operator, Adv. Math. 08,

19 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.9-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 9 log + D q fx 4 R q/ + q / x m R, f + m R, fx R x [R q / + q / x ] f + q / + q / x R x + R q / + q / x nr, f + nr, f + D α q / α + q / α log ασlog+ε x + 3 nr, f + nr, f + log 0 0 where D α : 4 C α 3 α, x / E. On the other hand, NR, f R R nr, f R nt, f n0, f dt + n0, f log R t 9 R R 9 Hence dt n0, f t Similarly, we have nr, f log R. ε We now choose α 33 σ log +ε and substitute x r, R r log r into Lemma 4. to 33 divided difference operator, Adv. Math. 08, R dt + n0, f log R t nr, f NR, f log R O[log R σlog+ ε log R O log σ log + ε R. ] n R, O log σ log + ε R. 60 f obtain the 7. We now compute the logarithmic measure of E. To do so, we first note the elementary inequality that given δ > 0 sufficiently small, there is a positive constant C δ so that log + t t C δt 6 for 0 t < δ. We assume, in the case when there are infinitely many {d n } otherwise, the logarithmic measure of E is obviously finite, they are ordered in the increasing moduli. Then we choose an N sufficiently large such that

20 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics log σlog+ε < δ. 6 d N Hence log-meas E dt t dt t + 6 E [, + E [, d N ] E [ d N, + 6 log d N + 9 nn nn log d N + C δ E n dt t log d N + log σ log+ε d n < nn π π dt t + /log σ log +ε d n log /log σlog+ε d n where the conclusion of the last sum is convergent follows from [6, Lemma 4.] for meromorphic functions of finite logarithm order σ log. Proof of Theorem 3.. We first prove a crucial estimate. Lemma 4.3. Let 0 < α < be given. Then for each fixed A C and an arbitrary w and we have re iφ Are iφ r e iφ w dφ E α α r α 63 for all r sufficiently large, where E α is independent of w. Here the square root is given by r e iφ re iφ as re iφ as the agreed convention in. Remark 4.4. We note that when A 0 in the above lemma, with the same 0 < α < and w C arbitrary, recovers the known estimate π holds for all r > 0 see e.g., [9, p. 6] and [3, p. 66]. re iφ w α dφ αr α, 64 Proof. Since for each r large enough, { } re iφ Are iφ r e iφ : 0 φ π is a closed curve enclosing the origin, therefore, for each non-zero w, the curve must intersect with the array { tw : 0 < t < } divided difference operator, Adv. Math. 08,

21 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P. -56 Y.-M. Chiang, S. Feng / Advances in Mathematics at least once. So we can find a real C > 0 and 0 β < π such that we can represent w as w C re iβ Are iβ r e iβ. 67 When w 0, we can still represent w by 67 with C 0. Substituting 67 into the integral on the left side of 63 yields 8 π 8 re iφ Are iφ r e iφ w dφ α π dφ re iφ Cre iβ A [ re iφ r e iφ Cre iβ r e iβ ] α 0 π dφ re iφ β Cr A [ re iφ β r e iφ β e iβ Cr r e iβ ] α π β 8 dφ re iφ Cr A [ re iφ r e iφ e iβ Cr r e iβ ] α 0 β 0 3π π π + π dφ re iφ Cr A [ re iφ r e iφ e iβ Cr r e iβ ] α : I + I. We now estimate I. We first consider re iφ Cr A [ re iφ r e iφ e iβ Cr r e iβ ] Cr Ar r e iβ 33 + re iφ Are iφ r + r e iβ r e iφ e iβ r Ar r e iβ C + reiφ Are iφ r + r e iβ r e iφ e iβ r Ar. r e iβ We see from the agreed convention that when and φ < π/ and r, 4 re iφ r e iφ e iβ e iβ 4 divided difference operator, Adv. Math. 08, re iφ + O/r 70 r e iφ + e iβ

22 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P. -56 Y.-M. Chiang, S. Feng / Advances in Mathematics and in particular, r r e iβ O/r, 7 where and henceforth in the rest of this proof the big-o notation denotes constants, not necessary the same each time, are independent of w and φ, though they may depend on the corresponding A. Then for all r sufficiently large, r Ar r e iβ r Since C is real, we deduce C + reiφ Are iφ r + r e iβ r e iφ e iβ r Ar r e iβ re iφ Are iφ r + r I e iβ r e iφ e iβ r Ar r e iβ R re iφ Are iφ r + r e iβ r e iφ e iβ I r Ar r e iβ + I re iφ Are iφ r + r e iβ r e iφ e iβ. R r Ar r e iβ We deduce from 7, and R r Ar r e iβ r + O r 3 74 I r Ar O r e iβ r 3 75 as r. Let us now estimate re iφ r + r e iβ r e iφ e iβ, which we rewrite into the form re iφ r + r e iβ r e iφ e iβ reiφ r r e iβ + r e iφ e iβ + r r e iφ r e iβ + 76 r e iφ e iβ divided difference operator, Adv. Math. 08,

23 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.3-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 3 re iφ r r r e iβ + re iφ r e iφ e iβ r e iβ + r e iφ e iβ r e iφ O/r r + re iφ, + O/r where we have applied the estimates 70 and 7 to both the numerator and denominator in the last step above. When φ < π and r sufficiently large r + re iφ + O/r r O /r r, 0 0 hence re iφ r + r e iβ r e iφ e iβ φ e iφ O/r O. r Therefore R re iφ Are iφ r + r e iβ r e iφ e iβ φ r cos φ + O Or φ, r and I re iφ Are iφ r + r e iβ r e iφ e iβ φ r sin φ + O r φ /πr φ + O r φ /. r Substituting 78, 79, 74 and 75 into 73 yields the inequality C + reiφ Are iφ r + r e iβ r e iφ e iβ r Ar r e iβ I re iφ Are iφ r + r e iβ r e iφ e iβ 33 R r Ar r e iβ R re iφ Are iφ r + r e iβ r e iφ e iβ I r Ar r e iβ divided difference operator, Adv. Math. 08,

24 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics r φ r + O r 3 Or φ O r 3 φ + O r φ 3, as r. We substitute 7 and 80 into 69 give re iφ Cr A [ re iφ r e iφ e iβ Cr r e iβ ] r φ /6 8 for r large enough. It follows from 8 that 0 0 I π π dφ re iφ Cr A [ re iφ r e iφ e iβ Cr r e iβ ] α π 6α 7 r α αr α 7 π φ α dφ α π α for all r sufficiently large. We can convert the integration range π, 3π of I into that of π, π by I π π π π π dφ re iφ+π Cr A [ re iφ+π r e iφ+π e iβ Cr r e iβ ] α π dφ re iφ Cr A [ re iφ + re iφ e iβ Cr r e iβ ] α dφ re iφ + Cr A [ re iφ re iφ e iβ + Cr r e iβ ] α 33 Notice that the integrand of this integral is the same as that of I with C replaced by C. We can easily see the above estimate for I is still valid when C is replaced by C. Hence I α π α αr α α 4α π 4 for all r sufficiently large. This completes the proof with E α : α divided difference operator, Adv. Math. 08,

25 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.5-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 5 Completion of the proof of Theorem 3.. Let us choose R r log r. Then we integrate the inequality 35 from 0 to π and apply the inequalities 63 and 64 to yield m r, D qfx O m r log r, f + m r log r, /f fx log r + O n r log r, f + n r log r, /f + O 8 π 8 + Or α re iφ c n α dφ 0 c n r log r 0 0 π + Or α x + cqq / z q / c n dφ α 3 c n r log r 3 0 π Or α dφ + O x cqq / z q / c n α c n r log r 0 O m r log r, f + m r log r, /f log r + O n r log r, f + n r log r, /f + O. As a result, we obtained the desired estimate 6 after applying 59 and Askey Wilson type counting functions and proof of Theorem 3.3 We need to set up some preliminary estimates first. Let g : C C be a map, not necessary entire. Let f be a meromorphic function on C and a Ĉ, we define the counting function nr, fgx a to be the number of a-points of f, counted according to multiplicity of f a at the point gx, in {gx : x < r}. The integrated counting function is defined by 3 r 3 N r, fgx a n t, fgx a n 0, fgx a n 0, fgx a log r. For z e iθ, we shall write 7 in the following notation where Dq f x : fq e iθ fq e iθ ĕq e iθ ĕq e iθ fˆx q fˇx q fˆx fˇx ˆx q ˇx q ˆx ˇx t dt divided difference operator, Adv. Math. 08,

26 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics ˆx ˆx q : q/ z + q / z, ˇx ˇx q : q / z + q / z. 87 Note that the maps ˆx and ˇx are analytic and invertible ˆˇx ˇˆx x when x is sufficiently large. The Theorem 3.3 is a direct consequence of the following Theorem. Theorem 5.. Let f be a meromorphic function of finite logarithmic order σ log f. Then for each extended complex number a Ĉ, and each ε > 0, we have 0 0 and similarly, N r, fˆx a N r, fx a + O log r σ log +ε + Olog r, 88 N r, fˇx a N r, fx a + O log r σ log +ε + Olog r, 89 N r, fˆx a N r, fx a + O log r σ log +ε + Olog r. 90 Here the meaning of Nr, fˆx a is interpreted as taking gx ˆx mentioned above. The expressions N r, fˇx a and N r, fˆx a have similar interpretations. Proof. We shall only prove the 88 since the 89 and 90 can be proved similarly. Let a µ µ N be a sequence of a-points of f, counting multiplicities. Recall that for x and hence z to be sufficiently large we have ˆˇx ˇˆx x. Therefore, there exists a sufficiently large M > and for r M, 5 r 5 N r, fˆx a n t, fˆx a n 0, fˆx a t 0 M 9 n t, fˆx a n 0, fˆx a r n t, fˆx a n 0, fˆx a 9 t dt M 3 + n 0, fˆx a log r n t, fˆx a dt + Olog r t 33 r M r 36 dt + n 0, fˆx a log r 38 M 38 n t, fˆx a n M, fˆx a log r + Olog r ǎ µ t dt + Olog r 40 M ǎ µ <r 40 by the definition 85. Then t dt divided difference operator, Adv. Math. 08,

27 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.7-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 7 N r, fx a N r, fˆx a log r + n0, fx a log r a µ 3 0< a µ <r M ǎ µ <r 3 log r a µ log r + Olog r ǎ µ 6 M a µ <r M ǎ µ <r 6 log r ǎ µ log r + a µ ǎ µ M a 9 µ <r, M ǎ µ <r, 9 M ǎ µ <r a µ r or a µ <M log r + Olog r a µ M a µ <r, ǎ µ r or ǎ µ <M log a µ + ǎ µ a µ 5 M ǎ µ <r, M ǎ µ <r, M a µ <r, 5 M a µ <r a µ r ǎ µ r Let us write where 33 ǎ 34 µ a µ a µ q / a µ 34 log r + Olog r ǎ µ divided difference operator, Adv. Math. 08, log r ǎ µ + log log r r + Olog r. 9 ˇx q / x + ηx, 9 ηx q/ q / x + x which clearly tends to zero as x. Thus, there exists a constant h > 0 such that 93 ηx h 94 for all sufficiently large x. Thus, it follows from Lemma 4. with α, 9 and 94 that for M ǎ µ, M a µ, Let log a µ log q / a µ + ηa µ log q / a µ + log q / a µ + ηa µ log q / + ηa µ q / a + ηa µ µ q / a µ + ηa µ log q / h + q / a µ + h ǎ µ. 95 c log q / + log q / + h q/ M + h q / M + h M. 96

28 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics Then, log a µ c 4 M ǎ µ <r, ǎ µ M ǎ µ <r, 4 5 M a µ <r M a µ <r 5 Similarly we have log r ǎ µ 0 M ǎ µ <r, M ǎ µ <r, M ǎ µ <r, 0 a µ r a µ r a µ r and log r a µ c 6 M a µ <r, M a µ <r, ǎ µ r ǎ µ r 6 Combining the 9, 97, 98 and 99 yields N r, fx a N r, fˆx a c M a µ <r M ǎ µ <r 33 a µ <r a µ < q / r 33 log a µ ǎ µ c + 97, For M ǎ µ < r and r large enough and taking into account of the 94, + Olog r. 00 a µ q / ǎ µ + ηǎ µ q / r. 0 This together with 00 and an inequality similar to 59 imply that, for every ε > 0, we have N r, fx a N r, fˆx a c + This completes the proof. + Olog r [ c n r, fx a + n q / r, fx a ] + Olog r O log r σ log +ε + Olog r. 0 The above estimate should be compared with the estimate of Nr, fx + η Nr, fx+or σ +ε obtained by the authors in [7, Theorem.] for a meromorphic function of finite order σ, where η is a fixed, though arbitrary non-zero, complex number. divided difference operator, Adv. Math. 08,

29 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.9-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 9 6. Proof of the Second Main Theorem 3.5 We shall follow Nevanlinna s argument 4 by replacing the f x by the AW-operator D q f [4, pp ]. The methods used in [7] and [5] were based on Mohon ko s theorem see [35, p. 9]. We let Fx : p. 03 fx A ν 8 ν 8 We deduce from [7, p. 5] that 0 0 mr, F m r, FD q f m r, D q f + m r, D q f 3 ν 3 8 ν 8 9 ν µ 9 5 µ 5 30 µ µ p D q f. 04 f A ν On the other hand, for a given µ amongst {,, p} we write F in the for Fx : + fx A µ p fx A µ. 05 fx A ν Let δ min{ A h A k, } whenever h k. We follow the argument used by Nevanlinna [4, pp ] to arrive at the inequality mr, F > Combining this inequality with 04 yields m r, > D q f p m r, A µ m r, p m p r, A µ p log log 3. δ p D q f p log p log f A µ δ Let us now add N r, /D q f on both sides of this inequality and utilizing the first main theorem [40] see also [7] and [4], we deduce Tr, D q f T r, + O m r, + N r, + O 07 D q f D q f D q f p > N r, + m p D q f r, A ν m r, + O. f A ν D q f 38 ν ν 38 But is it elementary that According to [4, pp ] Nevanlinna proved the original version of 3 for p 3 in 93 and the 4 4 general case for p > 3 for entire functions was due to Collingwood in 94 [9]. 4 divided difference operator, Adv. Math. 08,

30 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics Tr, D q f mr, D q f + Nr, D q f 08 Nr, D q f + m r, D q f/f + mr, f. Eliminating the Tr, D q f from the inequalities 08 and 07, adding mr, f on both sides of the combined inequalities and rearranging the terms yield mr, f + p mr, A ν Tr, f Nr, f Nr, D q f + N r, 9 ν 9 + m r, D qf + m r, f p D q f + O. f A ν 0 0 ν The inequality 3 now follows by noting the m r, p D q f m r, f A ν 7 ν ν ν 7 Theorem 3. and the 33. p D q f A ν f A ν 7. Askey Wilson type Second Main theorem part II: truncations D q f p m r, D qf A ν, f A ν We recall that in classical Nevanlinna theory, for each element a, the counting function n r, f a counts distinct a-points for a meromorphic function f in C can be written as a sum of integers h k summing over all the points x in { x < r} at which fx a with multiplicity h, and where k h is the multiplicity of f x 0 where fx a. We define an Askey Wilson analogue of the nr, f. We define the Askey Wilson-type counting function of f ñ AW r, f a ñ AW r, f a 0 to be the sum of integers of the form h k summing over all the points x in { x < r} at which fx a with multiplicity h, while the k is defined by k : min{h, k } and where k is the multiplicity of D q fˆx 0 at ˆx. Similarly, we define ñ AW r, f ñ AW r, f ñ AW r, f 0 to be the sum of integers h k, summing over all x in x < r at which /fx 0 with multiplicity h, k : min{h, k } and where k is the multiplicity of D q /fˆx 0 at ˆx. divided difference operator, Adv. Math. 08,

31 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P.3-56 Y.-M. Chiang, S. Feng / Advances in Mathematics 3 3 r and We define the Askey Wilson-type integrated counting function of fx by Ñ AW r, f a ÑAW 9 r 9 ñ 0 AW t, f ñ AW 0, f 0 Ñ AW r, f r, f a t ñ AW t, f a ñ AW 0, f a dt t + ñ AW 0, f a log r, 0 4 ν 4 33 dt + ñ AW 0, f log r. 3 The and 3 are respectively the analogues for the Nr, f a and Nr, f from the classical Nevanlinna theory. We are now ready to state an alternative Second Main Theorem in terms of the AW-type integrated counting function defined above. The theorem could be regarded as a truncated form of the original Second Main Theorem, the Theorem 3.5. Theorem 7.. Suppose that fz is a non-constant meromorphic function of finite logarithmic order σ log f as defined in 5 such that D q f 0, and let a, a,, a p where p, be mutually distinct elements in C. Then we have, for r < R and for every ε > 0, p Tr, f Ñ AW r, f + p Ñ AW r, f a ν + S log r, ε; f 4 where S log r, ε; f O log r σ log +ε +O log r holds for all x r sufficiently large, where ÑAW r, f a ν and ÑAW r, f are defined by and 3, respectively. This truncated form of the Second Main Theorem leads to new interpretation of Nevanlinna s original defect relation, deficiency, etc, and perhaps the most important of all, is a new type of Picard theorem gears toward the Askey Wilson operator. We will discuss the functions that lie in the kernel of the AW-operator in 0. We note that Halburd and Korhonen [4] was the first to give such a truncated form of a Second Main theorem for the difference operator fx fx + η fx. However, both the formulation of our counting functions ÑAWr and the method of proof differs greatly from their original argument. Proof of the Theorem 7.. We are ready to prove the Theorem 7.. Adding the sum Nr, f + p Nr, f a ν 5 4 ν 4 on both sides of 3 and rearranging the terms yields divided difference operator, Adv. Math. 08,

32 JID:YAIMA AID:655 /FLA [ml; v.3; Prn:/0/08; 9:8] P Y.-M. Chiang, S. Feng / Advances in Mathematics p Tr, f Nr, f + p Nr, f a ν N AW r, f + O log r σ log +ε 6 3 ν 3 holds for all x r > 0. It remains to compare the sizes of 5 and Ñ AW r, f + 8 ν 8 Subtracting 7 from 5 yields 0 0 Nr, f Ñ AW r, f + p Ñ AW r, f a ν. 7 p Nr, f aν ÑAW r, f a ν. 8 3 ν 3 It follows from the definitions of ñ AW r, f and ñ AW r, f a ν that the difference nr, f ñ AW r, f enumerates the number of zeros of D q /fˆx at which /fx has a zero in the disk x < r, with due count of multiplicities, while the difference nr, f a ν ñ AW r, f a ν enumerates the number of zeros of D q fˆx in the disk x < r at which fx a ν, with due count of multiplicities, and those points x in x < r that arise from the common a ν -points of fˆx and fˇx, respectively. We have fx f ˆx D q ˆx f ˆx x fx f ˆx D qfˆx. 9 fx f Recall that the maps ˆx and ˇx are analytic and invertible ˆˇx ˇˆx x when x is sufficiently large. It follows from 9 that the zeros of D q /fˆx originate from the poles of fx, f ˆx or from the zeros of D q fˆx. On the other hand, the poles of D q fˆx must be amongst the poles of fx and/or poles of f ˆx, and in this case, the multiplicity of zeros of D q /fˆx, which is non-negative, equals to subtracting the multiplicity of poles of D q fˆx from the sum of multiplicities of the poles of fx and the poles of f ˆx. It follows from this consideration and the definitions of 0 and that 33 Nr, f Ñ AW r, f + 35 ν 35 Nr, fx + N r, f ˆx + N r, p Nr, f aν ÑAW r, f a ν Nr, D q fˆx D q fˆx holds. We deduce from Theorem 5. and Theorem 3.4 that the followings 0 N r, fˆx N r, fx + O log r σ log +ε + Olog r; N r, D q fˆx N r, D q fx + O log r σ log +ε + Olog r; divided difference operator, Adv. Math. 08,