In this paper, we introduce a general and versatile difference operator for a meromorphic function f, with respect to two distinct meromorphic functions g and h, as follows: <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D_ {g,h}f\left( z \right) :=\frac{f\left( g\left( z \right) \right) -f\left( h\left( z \right) \right) }{g\left( z \right) -h\left( z \right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msub> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <mi>f</mi> <mfenced close=")" open="("> <mi>g</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mfenced> <mo>-</mo> <mi>f</mi> <mfenced close=")" open="("> <mi>h</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mfenced> </mrow> <mrow> <mi>g</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>-</mo> <mi>h</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. This operator not only unifies several classical difference operators including Hahn difference operator, Jackson q-difference operator, and the forward difference operator, but also serves to characterize a wide range of pivotal concepts such as even functions, periodic functions, and uniqueness polynomials. Furthermore, we investigate entire solutions to the Fermat-type functional equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f\left( z \right) ^m+\left( D_{g,h}f\left( z \right) \right) ^n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <msup> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mi>m</mi> </msup> <mo>+</mo> <msup> <mfenced close=")" open="("> <msub> <mi>D</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msub> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mfenced> <mi>n</mi> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where m, n are positive integers, g and h are distinct entire functions.

Bi-Composite Difference Operator and Entire Solutions of Fermat-type Functional Equation

期刊论文
发表日期：

2026年4月25日

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Yu Gong,
Liu Yang

摘要

In this paper, we introduce a general and versatile difference operator for a meromorphic function f, with respect to two distinct meromorphic functions g and h, as follows: $D_ {g,h}f\left( z \right) :=\frac{f\left( g\left( z \right) \right) -f\left( h\left( z \right) \right) }{g\left( z \right) -h\left( z \right) }$ $D_{g, h} f (z) : = \frac{f (g, (z)) - f (h, (z))}{g (z) - h (z)}$ . This operator not only unifies several classical difference operators including Hahn difference operator, Jackson q-difference operator, and the forward difference operator, but also serves to characterize a wide range of pivotal concepts such as even functions, periodic functions, and uniqueness polynomials. Furthermore, we investigate entire solutions to the Fermat-type functional equation $f\left( z \right) ^m+\left( D_{g,h}f\left( z \right) \right) ^n=1$ $f {(z)}^{m} + {(D_{g, h}, f, (z))}^{n} = 1$ , where m, n are positive integers, g and h are distinct entire functions.