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<p>This paper investigates a class of time scales for which the forward jump function is given by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1116_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma (t)=qt+h\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mi>t</mi> <mo>+</mo> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation>, where <i>q</i>, and <i>h</i> are constants. This framework allows us to treat the standard, <i>h</i>-, <i>q</i>-, and (<i>q</i>,&#xa0;<i>h</i>)-derivatives simultaneously as special cases of the delta derivative. We establish a key connection between the <i>n</i>th delta derivative and specific <i>n</i>th divided difference, which serves as the foundation for generalizing several classical results from <i>q</i>-calculus to the broader context of (<i>q</i>,&#xa0;<i>h</i>)-calculus. In the second part of the paper, we present explicit formulas for the <i>n</i>th delta derivative of a quotient of two functions, extending familiar results from classical calculus. As an application, we use the obtained results to study the (<i>q</i>,&#xa0;<i>h</i>)-analogs of the power and exponential functions, yielding explicit expressions for the <i>n</i>th derivatives of their reciprocals and leading to a novel <i>q</i>-binomial identity.</p>

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On (q, h)-differentiation: divided differences, quotient rules, and applications

  • 期刊论文

  • 发表日期：

    2025年8月12日

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  • Dragan S. Rakić

摘要

This paper investigates a class of time scales for which the forward jump function is given by \(\sigma (t)=qt+h\) σ ( t ) = q t + h , where q, and h are constants. This framework allows us to treat the standard, h-, q-, and (q, h)-derivatives simultaneously as special cases of the delta derivative. We establish a key connection between the nth delta derivative and specific nth divided difference, which serves as the foundation for generalizing several classical results from q-calculus to the broader context of (q, h)-calculus. In the second part of the paper, we present explicit formulas for the nth delta derivative of a quotient of two functions, extending familiar results from classical calculus. As an application, we use the obtained results to study the (q, h)-analogs of the power and exponential functions, yielding explicit expressions for the nth derivatives of their reciprocals and leading to a novel q-binomial identity.