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<p>This work is dedicated to Professor José Carlos Petronilho, renowned for his significant work in the area of Orthogonal Polynomials. Our focus here is to shed light on his contributions to quantum calculus theory. The first sections highlight the key ideas, developments, and applications, underscoring the lasting impact of his work. Furthermore, in the context of the general quantum calculus, by adding a new condition on the associated <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\,\beta \,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mi>β</mi> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation> function, we show that the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\,\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>-analogues of the exponential function, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\,\text{ e}_{p,\beta }(t)\,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mspace width="0.333333em" /> <msub> <mtext>e</mtext> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\,\text{ E}_{p,\beta }(t)\,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mspace width="0.333333em" /> <msub> <mtext>E</mtext> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation>, share similar properties with the classical exponential function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\,\text{ e}^t\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mspace width="0.333333em" /> <msup> <mtext>e</mtext> <mi>t</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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LATEST ADVANCEMENTS IN QUANTUM OPERATORS: A UNIFIED APPROACH

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • J.L. Cardoso

摘要

This work is dedicated to Professor José Carlos Petronilho, renowned for his significant work in the area of Orthogonal Polynomials. Our focus here is to shed light on his contributions to quantum calculus theory. The first sections highlight the key ideas, developments, and applications, underscoring the lasting impact of his work. Furthermore, in the context of the general quantum calculus, by adding a new condition on the associated \(\,\beta \,\) β function, we show that the \(\,\beta \) β -analogues of the exponential function, \(\,\text{ e}_{p,\beta }(t)\,\) e p , β ( t ) and \(\,\text{ E}_{p,\beta }(t)\,\) E p , β ( t ) , share similar properties with the classical exponential function \(\,\text{ e}^t\) e t .