On the infinite Borwein product raised to a positive real power

<jats:p>In this paper, we study properties of the coefficients appearing in the <jats:italic>q</jats:italic>-series expansion of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>∏</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>pn</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>δ</mml:mi> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, the infinite Borwein product for an arbitrary prime <jats:italic>p</jats:italic>, raised to an arbitrary positive real power <jats:inline-formula><jats:alternatives><jats:tex-math>$$\delta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>. We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$p=3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent <jats:inline-formula><jats:alternatives><jats:tex-math>$$\delta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the <jats:inline-formula><jats:alternatives><jats:tex-math>$$p=3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> case.</jats:p>

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