Infinite Product Expansion Research Articles

Binary theta functions and Borcherds products

We obtain infinite product expansions in the sense of Borcherds for theta functions associated with certain positive-definite binary quadratic and binary hermitian forms. Among other things, we show that every weight 1 binary theta function is a Borcherds product. In particular, binary theta functions have zeros only at quadratic irrationals.

A note on exotic integrals

We consider Bernoulli measures μ p \mu _p on the interval [ 0 , 1 ] [0,1] . For the standard Lebesgue measure the digits 0 0 and 1 1 in the binary representation of real numbers appear with an equal probability 1 / 2 1/2 . For the Bernoulli measures, the digits 0 0 and 1 1 appear with probabilities p p and 1 − p 1-p , respectively. We provide explicit expressions for various μ p \mu _p -integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of μ p \mu _p in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire functions, which admit explicit infinite product expansions and satisfy interesting properties, including connections with the Stirling numbers and the polylogarithm.

On expansions for nonlinear systems Error estimates and convergence issues

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied. First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation. Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input. Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion. Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

The 5-dissections of two infinite product expansions

In this paper, we establish 5-dissections of two infinite products and thereby prove that their q-series coefficients vanish for two arithmetic progressions. Moreover, our results also imply that the signs of coefficients of these infinite products are periodic with period 5 from $$n>1$$.

Some results on vanishing coefficients in infinite product expansions

Recently, M. D. Hirschhorn proved that, if $\sum_{n=0}^\infty a_nq^n := (-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$ and $\sum_{n=0}^\infty b_nq^n:=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$, then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. Motivated by the work of Hirschhorn, D. Tang proved some comparable results including the following: If $ \sum_{n=0}^\infty c_nq^n := (-q,-q^4;q^5)_\infty^3(q^3,q^7;q^{10})_\infty$ and $\sum_{n=0}^\infty d_nq^n := (-q^2,-q^3;q^5)_\infty^3(q,q^9;q^{10})_\infty$, then $c_{5n+3}=c_{5n+4}=0$ and $d_{5n+3}=d_{5n+4}=0$. In this paper, we prove that $a_{5n}=b_{5n+2}$, $a_{5n+1}=b_{5n+3}$, $a_{5n+2}=b_{5n+4}$, $a_{5n-1}=b_{5n+1}$, $c_{5n+3}=d_{5n+3}$, $c_{5n+4}=d_{5n+4}$, $c_{5n}=d_{5n}$, $c_{5n+2}=d_{5n+2}$, and $c_{5n+1}>d_{5n+1}$. We also record some other comparable results not listed by Tang.

VANISHING COEFFICIENTS IN FOUR QUOTIENTS OF INFINITE PRODUCT EXPANSIONS

Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged)40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$, where $a_{1}(n)$ is defined by $$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$

Infinite product exponents for modular forms

Recently, Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $$\Gamma _0(N)$$ Γ 0 ( N ) . Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $$X_0(N)$$ X 0 ( N ) is 0 or 1.

Generators of the ring of weakly holomorphic modular functions for $$\Gamma _1(N)$$ Γ 1 ( N )

For a positive integer N divisible by 4, 5, 6, 7 or 9, let \(\mathcal {O}_{1,N}(\mathbb {Q})\) be the ring of weakly holomorphic modular functions for the congruence subgroup \(\Gamma _1(N)\) with rational Fourier coefficients. We present explicit generators of the ring \(\mathcal {O}_{1,N}(\mathbb {Q})\) over \(\mathbb {Q}\) by making use of modular units which have infinite product expansions.

Infinite product expansion of the Fokker-Planck equation with steady-state solution.

We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

Structures for pairs of mock modular forms with the Zagier duality

Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on $\SL_2(\ZZ)$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called the Zagier duality. After the result of Zagier, this type duality was studied broadly in various view points including the theory of a mock modular form. In this paper, we consider this problem with the Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using holomorphic Poincare series and their supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $\xi_{-k}$ and quadric relations of the critical values of their $L$-functions.

FURTHER RESULTS ON VANISHING COEFFICIENTS IN INFINITE PRODUCT EXPANSIONS

AbstractWe extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if $$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$ for certain integers $k$, $m$, $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon [‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemporary Mathematics, 166 (American Mathematical Society, Providence, RI, 1994), 129–139], but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms of integer partitions.

Congruences for Ramanujan’s f and ω functions via generalized Borcherds products

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and {\omega} in detail.

Borcherds products on unitary groups

We provide a construction of the multiplicative Borcherds lift for unitary groups U(1,m), which takes weakly holomorphic elliptic modular forms and lifts them to meromorphic automorphic forms having infinite product expansions and taking their zeros and poles along Heegner divisors. The result is obtained through the transfer of Borcherds' theory to unitary groups via a suitable embedding of U(1,m) into O(2,2m). Further, we examine the values taken by the Borcherds products around the cusps of the symmetric domain of the unitary group. Finally, as an application of the lifting, we derive a modularity result for generating series with Heegner divisors as coefficients, along the lines of Borcherds' generalization of the Gross-Zagier-Kohnen theorem.

CONGRUENCE PROPERTIES OF BORCHERDS PRODUCT EXPONENTS

In his striking 1995 paper, Borcherds [Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math.120 (1995) 161–213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

Generalized moonshine, II: Borcherds products

The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster simple group. We introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half-planes. We recast this result using the Borcherds-Harvey-Moore singular theta lift, and show that the vector-valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps. For each element of the monster simple group, we construct an infinite dimensional Lie algebra, such that its denominator formula is an infinite product expansion of the automorphic function arising from that element's McKay-Thompson series. These Lie algebras have the unusual property that their simple roots and all root multiplicities are known. We show that under certain hypotheses, characters of groups acting on these Lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group.

G150012 環境特性を考慮した環境再現システム

In this paper, environment copying system is focused. It achieves saving and reproduction of real environment haptic sensation. In the skill trainings which involves consumption of the limited resource, the risk and cost are large. If environment copying system is used instead of limited resource, both the risk and cost of training become very little. Environment model is used in the environment copying system. The conventional environment model consists of a spring coefficient and a damper coefficient. But, the conventional environment model can not reproduce vibration of the real environment. The vibration order differs depending on the real environmental characteristic. In this paper, the environment model which vibration order can be changed is proposed. The environment model is based on wave equation. The environment model is simplified by infinite product expansion of hyperbolic function. The order of product is the order of vibration. The validity of proposed method is confirmed by experiment.

Factorizations that involve Ramanujan’s function k(q) = r(q)r 2(q 2)

In the “lost notebook”, Ramanujan recorded infinite product expansions for $$\frac{1} {{\sqrt r }} - \left( {\frac{{1 - \sqrt 5 }} {2}} \right)\sqrt r and \frac{1} {{\sqrt r }} - \left( {\frac{{1 + \sqrt 5 }} {2}} \right)\sqrt r ,$$ , where r = r(q) is the Rogers-Ramanujan continued fraction. We shall give analogues of these results that involve Ramanujan’s function k = k(q) = r(q)r2(q2).

TWISTED BORCHERDS PRODUCTS ON HILBERT MODULAR SURFACES AND THE REGULARIZED THETA LIFT

We construct a lifting from weakly holomorphic modular forms of weight 0 for SL 2(ℤ) with integral Fourier coefficients to meromorphic Hilbert modular forms of weight 0 for the full Hilbert modular group of a real quadratic number field with an infinite product expansion and a divisor given by a linear combination of twisted Hirzebruch–Zagier divisors. The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. We generalize the work by Bruinier and Yang who showed the existence of the lifting for prime discriminants using a different approach.

Eisenstein series and elliptic functions on [formula omitted

We generalize two identities involving Eisenstein series given in Chapter 19 of Ramanujanʼs second notebook. Four infinite families of Eisenstein series are obtained and their properties are investigated. The generating function of each infinite family is shown to be an elliptic function which has a simple infinite product expansion. Transformation properties of the Eisenstein series are investigated from first principles without using the transformation properties of Dedekindʼs eta-function.

Poincaré series and the divisors of modular forms

Recently, Bruinier, Kohnen and Ono obtained an explicit description of the action of the theta-operator on meromorphic modular formsffonSL2(Z)SL_2(\mathbb {Z})in terms of the values of modular functions at points in the divisor offf. Using this result, they studied the exponents in the infinite product expansion of a modular form and recurrence relations for Fourier coefficients of a modular form. In this paper, we extend these results to meromorphic modular forms onΓ0(N)\Gamma _0(N)for an arbitrary positive integerN>1N>1.