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In this paper, an Askey-Wilson version of the Wronskian-Casorati determinant $\mathcal{W}(f_{0}, \dots, f_{n})(x)$ for meromorphic functions $f_{0}, \dots, f_{n}$ is introduced to establish an Askey-Wilson version of the general form of the Second Main Theorem in projective space. This improves upon the original Second Main Theorem for the Askey-Wilson operator due to Chiang and Feng. In addition, by taking into account the number of irreducible components of hypersurfaces, an Askey-Wilson version of the Truncated Second Main Theorem for holomorphic curves into projective space with hypersurfaces located in $l$-subgeneral position is obtained., Comment: 50 pages

In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (T\^{o}hoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen., Comment: 40 pages. arXiv admin note: text overlap with arXiv:math/9310226 by other authors

We consider the first order $q$-difference equation \begin{equation}\tag{\dag} f(qz)^n=R(z,f), \end{equation} where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\not=1$, we show that, if $(\dag)$ has a zero order transcendental meromorphic solution, then $(\dag)$ reduces to a $q$-difference linear or Riccati equation, or to an equation that can be transformed to a $q$-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of $(\dag)$ are presented. Given that $(\dag)$ can be transformed into a difference equation, we proceed to discuss the growth of the composite function $f(\omega(z))$, where $\omega(z)$ is an entire function satisfying $\omega(z+1)=q\omega(z)$, and demonstrate how the proposed difference Painlev\'e property, as discussed in the literature, applies for $q$-difference equations., Comment: 13 pages

It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{\Delta_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $\Delta_c f=f(z+c)-f(z)$ and $a,c\in\mathbb{C}$. This estimate implies a lower bound for the Nevanlinna ramification term in terms of the difference operator with an arbitrary shift. As a consequence it follows, for instance, that if $f$ is an entire function of hyper-order $<1$ whose derivative does not attain a value $a\in\mathbb{C}$ often $$N\left(r,\frac{1}{f'-a}\right)=S(r,f),$$ then the finite difference $\Delta_c f$ cannot attain the value $ac$ significantly more often $$N\left(r,\frac{1}{\Delta_c f-ac}\right)=S(r,f).$$ Additional applications of the estimate above include a new type of a second main theorem, deficiency relations between $\Delta_cf$ and $f'$ and new Clunie and Mohon'ko type lemmas., Comment: 25 pages

A generalization of the second main theorem of tropical Nevanlinna theory is presented for noncontinuous piecewise linear functions and for tropical hypersurfaces without requiring a growth condition. The method of proof is novel and significantly more straightforward than previously known proofs. The tropical analogue of the Nevanlinna inverse problem is formulated and solved for tropical meromorphic functions and tropical hypersurfaces.

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form $f(z+1)^n=R(z,f)$ $(\dag)$ that have transcendental meromorphic solutions, where $R(z,f)$ is rational in both arguments. In this paper, we first complete the classification for the case $\deg_{f}(R(z,f))=n$ of~$(\dag)$ by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on~$(\dag)$. Then, we study the relations between $(\dag)$ and its differential counterpart $(f')^n=R(z,f)$. We show that most autonomous equations, singled out from~$(\dag)$ with $n=2$, have a natural continuum limit to either the differential Riccati equation $f'=a+f^2$ or the differential equation $(f')^2=a(f^2-\tau_1^2)(f^2-\tau_2^2)$, where $a\not=0$ and $\tau_i$ are constants such that $\tau_1^2\not=\tau_2^2$. The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method., Comment: 21 pages. arXiv admin note: text overlap with arXiv:2108.06085

It is shown that if the equation \begin{equation*} f(z+1)^n=R(z,f), \end{equation*} where $R(z,f)$ is rational in both arguments and $\deg_f(R(z,f))\not=n$, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term $R(z,f)$ takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case $\deg_f(R(z,f))=n$ of the equation above and thus provide a complete difference analogue of Steinmetz' generalization of Malmquist's theorem., Comment: 37 pages

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a $\delta$-subharmonic function is established allowing the case of hyper-order equal to one and minimal hyper-type, which improves the condition of the hyper-order less than one. Finally, we make a careful discussion of a well-known difference equation and give out the possible forms of the equation under a growth condition for the solutions., Comment: 36 pages. To appear in the Transactions of the American Mathematical Society: https://doi.org/10.1090/tran/8069

Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\log T(r,f)\leq r/(\log r)^{2+\nu}$, $\nu>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on difference quotients, where the exceptional set is of finite logarithmic measure. We show the necessity of this set by proving that it must be of infinite linear measure for meromorphic functions whose deficiency is dependent on the choice of the origin. In addition, we show that there is an infinite sequence of $r$ in the set for which $m(r,f(z+c)/f(z))$ is not small compared to $T(r,f)$ for entire functions constructed by Miles. We also give a discrete version of Borel type growth lemma and use it to extend Halburd's result on first order discrete equations of Malmuist type., Comment: 13 pages

Differential calculus is not a unique way to observe polynomial equations such as $a+b=c$. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials $a$, $b$ and $c$ satisfying the equation above. Then a difference $abc$ theorem for polynomials is proved using a new notion of a radical of a polynomial. Two results on the non-existence of polynomial solutions to difference Fermat type functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider difference Fermat type equations with transcendental entire solutions., Comment: 16 pages

It is shown that if It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag{\dag} \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic coefficients and $\deg_f(R(z,f))=n$, has an admissible meromorphic solution, then either $f$ satisfies a difference linear or Riccati equation with meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if \eqref{abstract_eq}, where the assumption $\deg_f(R(z,f))=n$ has been discarded, has rational coefficients and a transcendental meromorphic solution $f$ of hyper-order $<1$, then either $f$ satisfies a difference linear or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions which are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz' generalization of Malmquist's theorem., Comment: 25 pages, some errors are corrected

It is shown that the difference equation \begin{equation}\label{abseq} (\Delta f(z))^2=A(z)(f(z)f(z+1)-B(z)), \qquad\qquad (1) \end{equation} where $A(z)$ and $B(z)$ are meromorphic functions, possesses a continuous limit to the differential equation \begin{equation}\label{abseq2} (w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost same in the sense of Nevanlinna under some conditions. Examples are given to discuss the sharpness of the results obtained. These properties are counterparts of the corresponding results on the algebraic differential equation (2)., Comment: 12 pages

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations., Comment: 32 pages. This is the final version for being published. arXiv admin note: text overlap with arXiv:1601.05716

We extend the difference analogue of Cartan's second main theorem for the case of slowly moving periodic hyperplanes, and introduce two different natural ways to find a difference analogue of the truncated second main theorem. As applications, we obtain a new Picard type theorem and difference analogues of the deficiency relation for holomorphic curves.

Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations solved by elliptic functions.

Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of $f$ instead of its Wronskian determinant. As its applications, we obtain a defect relation, a uniqueness theorem and a difference analogue of generalized Picard theorem., Comment: 27 pages

Tropical Nevanlinna theory, introduced by Halburd and Southall as a tool to analyze integrability of ultra-discrete equations, studies the growth and complexity of continuous piecewise linear real functions. The purpose of this paper is to extend tropical Nevanlinna theory to n-dimensional tropical projective spaces by introducing a natural characteristic function for tropical holomorphic curves, and by proving a tropical analogue of Cartan's second main theorem. It is also shown that in the 1-dimensional case this result implies a known tropical second main theorem due to Laine and Tohge., Comment: 26 pages

Nevanlinna's second main theorem is a far-reaching generalisation of Picard's Theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f' appear. In this paper we show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalisations of Picard's Theorem and the defect relations., Comment: Accepted for publication in the Proceedings of the Edinburgh Mathematical Society

It is shown that if w(z) is a finite-order meromorphic solution of the equation H(z,w) P(z,w) = Q(z,w), where P(z,w) = P(z,w(z),w(z+c_1),...,w(z+c_n)) is a homogeneous difference polynomial with meromorphic coefficients, and H(z,w) = H(z,w(z)) and Q(z,w) = Q(z,w(z)) are polynomials in w(z) with meromorphic coefficients having no common factors such that max{deg_w(H), deg_w(Q) - deg_w(P)} > min{deg_w(P), ord_0(Q)} - ord_0(P), where ord_0(P) denotes the order of zero of P(z,x_0,x_1,...,x_n) at x_0=0 with respect to the variable x_0, then the Nevanlinna counting function N(r,w) satisfies N(r,w) > S(r,w). This implies that w(z) has a relatively large number of poles. For a smaller class of equations a stronger assertion N(r,w) = T(r,w)+S(r,w) is obtained, which means that the pole density of w(z) is essentially as high as the growth of w(z) allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painleve type., Comment: 12 pages

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if three distinct values of a meromorphic function f of hyper-order less than 1/n^2 have forward invariant pre-images with respect to a fixed branch of the algebraic function t(z)=z+a_{n-1} z^{1-1/n}+...+a_1 z^{1/n}+a_0 with constant coefficients, then f(t(z)) = f(z) for all z. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images., Comment: 15 pages, 1 figure. Typographical errors corrected in v2

It is shown that if three distinct values of a meromorphic function f:C^n -> P^1 of hyper-order strictly less than 2/3 have forward invariant pre-images with respect to a translation t:C^n -> C^n, t(z)=z+c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green's Picard-type theorem in the special case where the hyper-order of f is less than 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the lemma on the logarithmic derivative and of the second main theorem of Nevanlinna theory for meromorphic functions C^n -> P^1 are given, and their applications to partial difference equations are discussed., Comment: 19 pages

If f: C -> P^n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation t(z)=z+c, then f is periodic with period c. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples., Comment: 32 pages, to appear in Trans. Amer. Math. Soc

Sharp versions of some classical results in differential equations are given. Main results consists of a Clunie and a Mohon'ko type theorems, both with sharp forms of error terms. The sharpness of these results is discussed and some applications to nonlinear differential equations are given in the conluding remarks. Moreover, a short introduction on the connection between Nevanlinna theory and number theory, as well as on their relation to differential equations, is given. In addition, a brief review on the recent developments in the field of sharp error term analysis is presented., Comment: 16 pages, contains an erratum at the end of the paper