Ashoka University is a private university located in Sonipat, Haryana, India, focusing on liberal arts and sciences. It was established in 2014.&nbsp;It is guided by an academic council of academics and scholars from around the world. The council sets the university&#39;s academic standard, advising on matters of curriculum, faculty hiring and research.&nbsp;

Ashoka University offers a full-time 3 (+ 1) year undergraduate programme. The University comprises of&nbsp;2500 plus students, with a mix of both national and international students.&nbsp; &nbsp;

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Ashoka University

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite number-theoretic matrix. We call this matrix the partition-frequency enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-theoretic functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan’s 𝜏<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi></math>&nbsp;function, sums of squares and triangular numbers, and for 𝜁(2𝑛)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ζ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></math>, where n is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer, and others. As one application, we embed Ramanujan’s famous congruences 𝑝(5𝑛+4)≡0(mod5)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">(</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≡</mo><mn>0</mn><mspace width="thickmathspace">&nbsp;</mspace><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="thinmathspace">&nbsp;</mspace><mn>5</mn><mo stretchy="false">)</mo></math>&nbsp;and 𝜏(5𝑛+5)≡0(mod5)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>≡</mo><mn>0</mn><mspace width="thickmathspace">&nbsp;</mspace><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="thinmathspace">&nbsp;</mspace><mn>5</mn><mo stretchy="false">)</mo></math>&nbsp;into an infinite family of such congruences.

Department of Mathematics

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Fulltext

The Ramanujan Journal

The partition-frequency enumeration matrix

We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using q-Lagrange inversion. We use the&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>,&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math>&nbsp;basic <a href="https://www.sciencedirect.com/topics/mathematics/hypergeometric-series">hypergeometric series</a>.