Holomorphic curves with shift-invariant hyperplane preimages

Halburd, R. G.; Korhonen, Risto; Tohge, Kazuya

doi:10.1090/s0002-9947-2014-05949-7

Cited by 183 publications

(140 citation statements)

References 34 publications

(47 reference statements)

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“…We proceed to consider the situation for q-difference equations recalling first the definition of a q-Casorati determinant (see [16]). If f 1 , .…”

Section: Q-casorati Determinantsmentioning

confidence: 99%

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FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

Wen¹

2014

Bulletin of the Korean Mathematical Society

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Abstract. During the last decade, several papers have focused on linear q-difference equations of the formwith entire or meromorphic coefficients. A tool for studying these equations is a q-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order ρ log . It is shown, under certain assumptions, that ρ log (f ) = max{ρ log (a j )} + 1. Moreover, it is illustrated that a q-Casorati determinant plays a similar role in the theory of linear q-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the q-gamma function and the q-exponential functions all have logarithmic order two.

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“…We proceed to consider the situation for q-difference equations recalling first the definition of a q-Casorati determinant (see [16]). If f 1 , .…”

Section: Q-casorati Determinantsmentioning

confidence: 99%

“…Since T (r, f ) is increasing, it is now not difficult to see that the last equation also holds if r → ∞ through any sequence of rvalues. If |q| < 1, we denote s = 1/q and make a change of a variable in (16) by replacing z with s n z. This leads to a linear s-difference equation, corresponding to (16).…”

Section: Lemma 52 ([19] Lemma 33)mentioning

confidence: 99%

FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

Wen¹

2014

Bulletin of the Korean Mathematical Society

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“…Theorem 4.1 implies Nevanlinna's second main theorem in the complex plane by our choosing L(f ) = f ′ and N = M. By choosing L(f ) = ∆(f ) = f (z + 1) − f (z) and N to be the field of meromorphic functions of hyper-order strictly less than one, Theorem 4.1 reduces into the difference analogue of the second main theorem [6,5]. Similarly, with the choice L(f ) = f (qz) − f (z), where q ∈ C \ {0, 1}, and taking N to be the field of zero-order meromorphic functions, we obtain the q-difference version of the second main theorem [2].…”

Section: General Linear Operatormentioning

confidence: 99%

“…For instance, if L(f ) = f ′ (z + 1), where the hyper-order (or iterated 2-order) ς(f ) of f satisfies ς(f ) = ς < 1, then by the lemma on the logarithmic derivative and its difference analogue [5], it follows that…”

Section: General Linear Operatormentioning

confidence: 99%

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Value Distribution and Linear Operators

Halburd

Korhonen

2013

Proceedings of the Edinburgh Mathematical Society

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Abstract. 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. IntroductionNevanlinna theory studies the value distribution of meromorphic functions. Central to the classical theory is the second main theorem together with the related defect relations, which are powerful generalisations of Picard's Theorem. Nevanlinna's second main theorem uses the distribution of points in the closed disc |z| ≤ r at which a meromorphic function f takes certain prescribed values to bound the Nevanlinna characteristic T (r, f ). The theorem incorporates information from the distribution of zeros and poles of the derivative f ′ through a ramification term which ensures that when we count the preimages of the prescribed values we can ignore multiplicities.In this paper we derive an analogue of the second main theorem in which the derivative f → f ′ is replaced by an arbitrary linear operator f → L(f ) on any subfield N of the space of meromorphic functions such that m(r, L(f )/f ) = o(T (r, f )) for r in a large subset of (0, ∞). Examples of such operators are the derivative f (z) → f ′ (z), the shift f (z) → f (z + c) and the q-difference operator f (z) → f (qz) as well as combinations such asWe derive a generalisation of Picard's Theorem, which in essence says that if there are enough functions a j ∈ ker(L) that are small compared with f , then L(f ) is identically zero. Analogues of the defect relations are given and several examples are used to illustrate the strength of the results obtained.The work presented here extends earlier work on the shift [6] and q-difference [2] operators. Those works in turn grew out of a programme to use Nevanlinna theory as a tool for detecting and describing difference equations of "Painlevé type" [1,7]. The generalisations described in the present paper are motivated by preliminary studies of differential delay equations such as αf (z) + βf ′ (z) = f (z) [f (z + 1) − f (z − 1)] , (1.1)2000 Mathematics Subject Classification. Primary 30D35, Secondary 34M03, 34M05, 39A06.

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Difference analogues of the second main theorem for meromorphic functions in several complex variables

Cao

2013

Mathematische Nachrichten

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In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard‐type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137, 457–474 (2007); Comput. Meth. Funct. Theor. 12, No. 1, 343–361 (2012)]. The method we used is very different from theirs.

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Holomorphic curves with shift-invariant hyperplane preimages

Cited by 183 publications

References 34 publications

FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

Value Distribution and Linear Operators

Difference analogues of the second main theorem for meromorphic functions in several complex variables

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