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<p>The purpose of this article is to continue our studies of single and multiple (<i>q</i>-)hypergeometric functions. We shall thus export the so-called <i>q</i>-case of multiple hypergeometric functions, Appell’s transformation formula, first Lauricella function transformation formula, Euler-Pfaff formulas for triple functions, transformation formula between the first and third Appell functions, integral representation and difference equations. First we prove Euler-Pfaff and reduction formulas for triple <i>q</i>-Saran hypergeometric functions including an equivalence relation for them. Then we shall prove both <i>q</i>-Euler and <i>q</i>-Laplace integral representations, as well as formal <i>q</i>-integral representations with the third <i>q</i>-real number. As usual, <i>q</i>-Euler integral representations are proved by the <i>q</i>-Beta integral. The <i>q</i>-Laplace integral representations contain confluent <i>q</i>-hypergeometric functions, which were previously defined. These formulas are proved by using the <i>q</i>-integral expression for the <i>q</i>-Gamma function. Because of the confluence, powers of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((1-q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> occur in several formulas and two new triple, confluent <i>q</i>-hypergeometric functions are used. Furthermore, systems of <i>q</i>-difference equations for some <i>q</i>-Saran functions are stated whose proofs are obvious. Some of Sarans formulas are corrected, and in some cases new triple hypergeometric formulas are inserted. Finally, two transformations for the first <i>q</i>-Lauricella function are proved, one of which requires the use of a <i>q</i>-real number.</p>

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On Transformations and q-Integrals for Multiple (q-)Hypergeometric Series

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Thomas Ernst

摘要

The purpose of this article is to continue our studies of single and multiple (q-)hypergeometric functions. We shall thus export the so-called q-case of multiple hypergeometric functions, Appell’s transformation formula, first Lauricella function transformation formula, Euler-Pfaff formulas for triple functions, transformation formula between the first and third Appell functions, integral representation and difference equations. First we prove Euler-Pfaff and reduction formulas for triple q-Saran hypergeometric functions including an equivalence relation for them. Then we shall prove both q-Euler and q-Laplace integral representations, as well as formal q-integral representations with the third q-real number. As usual, q-Euler integral representations are proved by the q-Beta integral. The q-Laplace integral representations contain confluent q-hypergeometric functions, which were previously defined. These formulas are proved by using the q-integral expression for the q-Gamma function. Because of the confluence, powers of \((1-q)\) ( 1 - q ) occur in several formulas and two new triple, confluent q-hypergeometric functions are used. Furthermore, systems of q-difference equations for some q-Saran functions are stated whose proofs are obvious. Some of Sarans formulas are corrected, and in some cases new triple hypergeometric formulas are inserted. Finally, two transformations for the first q-Lauricella function are proved, one of which requires the use of a q-real number.