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In this work, we give an elementary proof of the transformation formula for the Dedekind eta function under the action of the modular group PSL (2;Z). We start by giving a proof of the transformation formula η(τ) under the transformation τ →-1/τ, using the Jacobi triple product identity and the Poisson summation formula. After we establish some identities for the Dedekind sum, the transformation formula for η(τ) under the transformation induced by a general element of the modular group PSL (2;Z) is derived by induction. [ABSTRACT FROM AUTHOR]

Srinivasa Ramanujan developed seventeen fast convergent series for 1/π. Motivated by Ramanujan's series for 1/π many mathematicians have developed many theories for deriving new series for 1/π. In this article we obtain new series for 1/π using Eisenstein series of level three. [ABSTRACT FROM AUTHOR]

The zeros of the nth D'Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind η -function. In this paper, we prove that these coefficients are non-vanishing for | x | > κ (n - 1) and κ ≈ 9.7225 . Numerical computations imply that 9.72245 is a lower bound for κ . This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions. [ABSTRACT FROM AUTHOR]

The zeros of D'Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D'Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials. [ABSTRACT FROM AUTHOR]

For | q | < 1 , define f i = ∏ n = 1 ∞ (1 − q i n) , and let (A (q) , B (q)) be any of the pairs (f 1 4 , f 1 8 f 2 2 ) , (f 1 4 , f 1 1 0 f 3 2 ) , (f 1 6 , f 2 4 f 1 2 ) , (f 1 6 , f 1 1 4 f 2 4 ) , (f 1 1 0 , f 2 6 f 1 2 ) , (f 1 1 4 , f 3 5 f 1 ) , (f 1 1 4 , f 2 8 f 1 2 ). For any such pair (A (q) , B (q)) , define the sequences { a (n) } and { b (n) } to be the coefficients of q n of A (q) and B (q) , respectively. Then for each pair it is shown that a (n) vanishes if and only if b (n) vanishes. In each case, a criterion is given which states precisely when a (n) = b (n) = 0. Moreover, for the pairs (f 1 2 6 , f 3 9 f 1 ) , (f 1 2 6 , f 2 1 6 f 1 6 ) it is shown that a (n) = b (n) = 0 if 1 2 n + 1 3 satisfies a criteria of Serre for a (n) = 0. [ABSTRACT FROM AUTHOR]

We study a type of generalized double coset operators which may change the characters of modular forms. For any pair of characters v 1 and v 2 , we describe explicitly those operators mapping modular forms of character v 1 to those of v 2. We give three applications, concerned with eta-quotients. For the first application, we give many pairs of eta-quotients of small weights and levels, such that there are operators map one eta-quotient to another. We also find out these operators. For the second application, we apply the operators to eta-powers whose exponents are positive integers not greater than 24. This results in recursive formulas of the coefficients of these functions, generalizing Newman's theorem. For the third application, we describe a criterion and an algorithm of whether and how an eta-power of arbitrary integral exponent can be expressed as a linear combination of certain eta-quotients. [ABSTRACT FROM AUTHOR]

We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form. [ABSTRACT FROM AUTHOR]

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun. As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series. [ABSTRACT FROM AUTHOR]

In this article, we have evaluated the convolution sums ∑ a l + b m = n σ (l) σ (m) for a · b = 15 , where a , b ∈ N , using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [24]. As a consequence, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form x 1 2 + x 1 x 2 + x 2 2 + x 3 2 + x 3 x 4 + x 4 2 + 5 x 5 2 + x 5 x 6 + x 6 2 + x 7 2 + x 7 x 8 + x 8 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in ∏ n = 1 ∞ 1 + 1 F n , ∏ n = 3 ∞ 1 - 1 F n , ∏ n = 1 ∞ 1 + 1 L n , ∏ n = 2 ∞ 1 - 1 L n are algebraically independent over Q , where { F n } and { L n } are the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 and N be positive integers and let χ be a Dirichlet character modulo N. Let f(z) be a modular form in M k (Γ 0 (N) , χ) . Then we have a unique decomposition f (z) = E f (z) + S f (z) , where E f (z) ∈ E k (Γ 0 (N) , χ) and S f (z) ∈ S k (Γ 0 (N) , χ) . In this paper, we give an explicit formula for E f (z) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities. [ABSTRACT FROM AUTHOR]

In this paper we show some identities come from the q-identities of Euler, Jacobi, Gauss and Rogers-Ramanujan. Some of these identities relating the function sum of divisor of a positive integer n and the number of integer partitions of n. One of the most intriguing result found here is given by the next equation, for n > 0. ..., where σ1(n) is the sum of all positive divisors of n, p1(n) is the number of integer partitions of n, and C(n) is the set of integer compositions of n. In the last section we show seven applications, one of them is a series expansion for ..., where a1, ..., ak, b1, ..., bk, c1, ..., cr, d1, ... dr are positive integers, and |q| < 1. [ABSTRACT FROM AUTHOR]

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and 0 < h (n) ≤ h (n + 1) . We put P 0 g , h (x) = 1 and P n g , h (x) : = x h (n) ∑ k = 1 n g (k) P n - k g , h (x). As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind η -function. Here, g is the sum of divisors and h the identity function. Kostant's result on the representation of simple complex Lie algebras and Han's results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein's j-invariant, and Chebyshev polynomials of the second kind. [ABSTRACT FROM AUTHOR]

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace–Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker's limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori. [ABSTRACT FROM AUTHOR]

In this work we study the determinant of the Laplace–Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments. Refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved. [ABSTRACT FROM AUTHOR]

For a positive integer N , let C (N) be the subgroup of J 0 (N) generated by the equivalence classes of cuspidal divisors of degree 0 and C (N) (Q) : = C (N) ∩ J 0 (N) (Q) be its Q -rational subgroup. Let also C Q (N) be the subgroup of C (N) (Q) generated by Q -rational cuspidal divisors. We prove that when N = n 2 M for some integer n dividing 24 and some squarefree integer M , the two groups C (N) (Q) and C Q (N) are equal. To achieve this, we show that all modular units on X 0 (N) on such N are products of functions of the form η (m τ + k / h) , m h 2 | N and k ∈ Z and determine the necessary and sufficient conditions for products of such functions to be modular units on X 0 (N). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present some examples of positive-definite integral nondiagonal quaternary quadratic forms whose representation numbers can be determined explicitly using the theory of modular forms. Very few such examples appear in the literature. The seven forms presented were selected because they each belong to a genus containing exactly two form classes for which the single genus mate is a diagonal form whose representation number has been determined recently. [ABSTRACT FROM AUTHOR]

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$ -dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty)$ , initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms. [ABSTRACT FROM AUTHOR]

Kostant proved that the mth coefficient of the (r 2 - 1) th power of the Dedekind eta function is non-vanishing for r ≥ max { 4 , m } and r an integer. This was generalized by Han to the real case applying the Nekrasov–Okounkov hook length formula. We extend these results to the complex case and identify coefficients of the zth power with values of polynomials P m (- z) . We assume that all non-trivial roots have a negative real part. Numerical data indicate that the quadratic bound can be replaced by a linear bound. Han showed by a reversion method that P m m + 1 is divisible by m + 1 . We extend this result to P m ± n , where m and n are two coprime positive integers. We obtain congruences modulo n in the case P p ± n , where p is a prime. This leads to new non-vanishing results for the pth power of the Dedekind eta function, not covered in the theorems of Kostant and Han. We illustrate our results with numerical data and examples. [ABSTRACT FROM AUTHOR]

Two level 17 modular functions r = q 2 ∏ n = 1 ∞ (1 - q n) n 17 , s = q 2 ∏ n = 1 ∞ (1 - q 17 n) 3 (1 - q n) 3 are used to construct a new class of Ramanujan–Sato series for 1 / π . The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan's Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived. [ABSTRACT FROM AUTHOR]

Let g, h be two arithmetic functions. In this paper, we study zero-free intervals of polynomials of the type P n g , h (x) := x h (n) ∑ k = 1 n g (k) P n − k g , h (x) , P 0 g , h (x) = 1. We use a difference equation approach. Let g (n) = σ (n) be the divisor sum function and h (n) the identity function, then the zeros of the polynomials dictate the vanishing properties of the coefficients of all powers of the Dedekind eta function. This is directly related to the Lehmer conjecture on the non-vanishing of the Ramanujan τ-function. Let g (n) be the identity function. Then, the polynomials are related to generalized Laguerre polynomials for h (n) = n and to Chebyshev polynomials of the second kind for h (n) = 1. The paper is furnished with several interesting examples. We found polynomials with decreasing minimal real zeros. We provide a simple arithmetic function g to approximate the minimal real zeros attached to σ. [ABSTRACT FROM AUTHOR]

It is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2 m + 1) and ζ(2 Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2 m + 1 − 1/ N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η (z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ. [ABSTRACT FROM AUTHOR]

D'Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend results of Kostant and Han. [ABSTRACT FROM AUTHOR]

Let L ∈ { 6 , 9 , 12 } . We determine the generating functions of certain combinations of three ranks and three cranks modulo L in terms of eta quotients. Then, using the periodicity of signs of these eta quotients, we compare their values with the values of p (n) L / 3 . [ABSTRACT FROM AUTHOR]

The convolution sum, ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for ζ (2 m + 1). The formula for ζ 2 (2 m + 1) is then generalized in two different directions, one, by considering the generalized divisor function σ z (n) , and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter N. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for ζ (1 + ω) ζ (− 1 − ω) , where ω is a non-trivial zero of ζ (z). We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of ζ (4 k − 1) and ζ (4 k + 1) for k ∈ N. [ABSTRACT FROM AUTHOR]

We evaluate the convolution sum Wa,b (n) := ∑al+bm=n σ9(I)σ(m) for (a,b) = (1,28), (4,7), (2, 7), for all positive integers n. We use a modular form approach. We also re-evaluate the known sums W1,14(n) and W1,7(n) with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form x 1²+2²+3²+4²+7(5²+6²+7²+8²). Finally we express the modular forms ∆4,7(z), ∆4,14,1(z) and ∆4,14,2(z) (given in [10, 14]) as linear combinations of eta quotients. [ABSTRACT FROM AUTHOR]

Families of polynomials associated to arithmetic functions g (n) are studied. The case g (n) = σ (n) , the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials P n g (X) are defined by n -term recurrence relations. For the case that g (x) is a polynomial of degree d , we prove that at most a d + 2 term recurrence relation is needed. For the special case g (x) = x , we obtain explicit formulas and results. [ABSTRACT FROM AUTHOR]

We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the convolution sums ∑ a + p b = n σ (a) σ (b) , ∑ p 1 a + p 2 b = n σ (a) σ (b) and ∑ a + p 1 p 2 b = n σ (a) σ (b) where p , p 1 , p 2 are primes. [ABSTRACT FROM AUTHOR]

We find formulas for convolutions of sum of divisor functions twisted by the Dirichlet character − 4 ∗ , which are analogous to Ramanujan's formula for convolution of usual sum of divisor functions. We use the theory of modular forms to prove our results. [ABSTRACT FROM AUTHOR]

Abstract Let k , N ∈ N with N square-free and k > 1. We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any f (z) ∈ M 2 k (Γ 0 (N)) in terms of sum of divisors function. In particular, if f (z) ∈ E 2 k (Γ 0 (N)) , then the computation will to yield to an expression for the Fourier coefficients of f (z). Then we apply our main theorem to give formulas for convolution sums of the divisor function to extend the result by Ramanujan, and to eta quotients which yields to formulas for number of representations of integers by certain families of quadratic forms. At last we give essential results to derive similar results for modular forms in a more general setting. [ABSTRACT FROM AUTHOR]

The Fourier coefficients of powers of the Dedekind eta function can be studied simultaneously. The vanishing of the coefficients varies from super lacunary (Euler, Jacobi identities) and lacunary (CM forms) to non-vanishing (Lehmer conjecture for the Ramanujan numbers). We study polynomials of degree n, whose roots control the vanishing of the nth Fourier coefficients of such powers. We prove that every root of unity appearing as any root of these polynomials has to be of order 2. [ABSTRACT FROM AUTHOR]

We show that for any positive integer N, there are only finitely many holomorphic eta quotients of level N, none of which is a product of two holomorphic eta quotients other than 1 and itself. This result is an analog of Zagier's conjecture/Mersmann's theorem which states that of any given weight, there are only finitely many irreducible holomorphic eta quotients, none of which is an integral rescaling of another eta quotient. We construct such eta quotients for all cubefree levels. In particular, our construction demonstrates the existence of irreducible holomorphic eta quotients of arbitrarily large weights. [ABSTRACT FROM AUTHOR]

We find all the eta quotients in the spaces M1 (Γ0(12), (./d))(d = -3, -4) of modular forms and determine their Fourier coefficients, where (./d) is the Legendre-Jacobi-Kronecker symbol. [ABSTRACT FROM AUTHOR]

In this paper, we study the vanishing properties of Fourier coefficients of rth powers of the Dedekind eta function. We give a certain type of classification of this property based on r. Further we extend the results of Atkin, Cohen, and Newman for odd powers. This leads to a partial answer to a question raised by van Lint. We also indicate possible generalization of the Lehmer conjecture. [ABSTRACT FROM AUTHOR]

Abstract In this article, we define functions analogous to Ramanujan's function f (n) defined in his famous paper "Modular equations and approximations to π ". We then use these new functions to study Ramanujan's series for 1 / π associated with the classical, cubic and quartic bases. [ABSTRACT FROM AUTHOR]

In this paper, we aim to prove that the Fourier coefficients of the modular function η 2 4 (τ) η 2 4 (2 τ) possess a sign-change property. [ABSTRACT FROM AUTHOR]

In a well known 1914 paper, Ramanujan gave a number of rapidly converging series for 1/π which are derived using modular functions of higher level. Chudnovsky and Chudnovsky in their 1988 paper derived an analogous series representing 1/π using the modular function J of level 1, which results in highly convergent series for 1/π, often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard-Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard-Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant J of level 1, we determine all Chudnovsky-Ramanujan type formulae which are valid around one of the three singular points: 0,1,∞. [ABSTRACT FROM AUTHOR]

Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson series of level 20. These equations induce expansions for modular forms of level 20 in terms of modular functions. The theory of singular values is applied to derive expansions for 1 / π of level 20 analogous to those formulated by Ramanujan. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm which can prove algebraic relations involving η -quotients, where η is the Dedekind eta function. [ABSTRACT FROM AUTHOR]

A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for | q | < 1 {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that ∫ 0 e - π ∏ n = 1 ∞ ( 1 - q 2 ⁢ n ) 20 ( 1 - q n ) 16 ⁢ d ⁢ q = 1 16. \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals. [ABSTRACT FROM AUTHOR]

We use the theory of modular forms to give formulas for the number of representations of n by all sextenary quadratic forms with coefficients 1 , 2 , 3 , 6 . We also apply our results to write newforms in S 3 ( Γ 0 ( 24 ) , χ ) in terms of eta quotients. [ABSTRACT FROM AUTHOR]

We give a short proof of Zagier's conjecture/Mersmann's theorem which states that each holomorphic eta quotient of weight 1/2 is an integral rescaling of some eta quotient from Zagier's list of fourteen primitive holomorphic eta quotients. In particular, given any holomorphic eta quotient f of weight 1/2, this result enables us to provide a closed-form expression for the coefficient of q n in the q -series expansion of f , for all n . We also demonstrate another application of the above theorem in extending the levels of the simple (resp. irreducible) holomorphic eta quotients. [ABSTRACT FROM AUTHOR]