Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann T...Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results and the consequences assuming RH are true.展开更多
1 Introduction Jacobi forms are the generalization of Jacobi theta series and the coefficients of the Fourier-Jacobi expansion of a Siegel modular form. The theory develops systematically in recent years and has many ...1 Introduction Jacobi forms are the generalization of Jacobi theta series and the coefficients of the Fourier-Jacobi expansion of a Siegel modular form. The theory develops systematically in recent years and has many interesting applications in theory of modular forms and number theory.展开更多
In this review article, we revisit derivation of the cumulative density function (CDF) of the test statistic of the one-sample Kolmogorov-Smirnov test. Even though several such proofs already exist, they often leave o...In this review article, we revisit derivation of the cumulative density function (CDF) of the test statistic of the one-sample Kolmogorov-Smirnov test. Even though several such proofs already exist, they often leave out essential details necessary for proper understanding of the individual steps. Our goal is filling in these gaps, to make our presentation accessible to advanced undergraduates. We also propose a simple formula capable of approximating the exact distribution to a sufficient accuracy for any practical sample size.展开更多
In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta funct...In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.展开更多
By Witten rigidity theorem and the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, the elliptic genus of a homogeneous spin manifold G/H can be expressed as a sum of theta functions quotients over the Weyl gro...By Witten rigidity theorem and the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, the elliptic genus of a homogeneous spin manifold G/H can be expressed as a sum of theta functions quotients over the Weyl group of G. Consequently, we obtainseveral classes of combinatorial identities of theta functions.展开更多
We shall study the differential equation y^l2=Tn（y）-（1-2μ2）;where μ2 is a constant, Tn（x） are the Chebyshev polynomials with n = 3,4,6. The solutions of the differential equations will be expressed explicitly...We shall study the differential equation y^l2=Tn（y）-（1-2μ2）;where μ2 is a constant, Tn（x） are the Chebyshev polynomials with n = 3,4,6. The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on 2F1 （1/4, 3/4; 1; z）, 2F1 （l/3, 2/3; 1; z）, 2F1 （1/6, 5/6; 1; z） and provide a unified approach to a set of identities of Rmanujan involving these hypergeometric functions.展开更多
Let G be the group of the fractional linear transformations generated by T（τ）=τ ＋ λ, S（τ）=（τ cos π/n ＋ sin π/n）/（-τ sin π/n ＋ cos π/n）;where λ=2（cos π/m ＋ cos π/n）/sin π/n;m, n is a pair of...Let G be the group of the fractional linear transformations generated by T（τ）=τ ＋ λ, S（τ）=（τ cos π/n ＋ sin π/n）/（-τ sin π/n ＋ cos π/n）;where λ=2（cos π/m ＋ cos π/n）/sin π/n;m, n is a pair of integers with either n ≥ 2, m ≥ 3 or n ≥ 3, m ≥ 2; τ lies in the upper half plane H.A fundamental set of functions f0, fi and f∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan＇s triple differential equations associated with the group G and establish the connection of f0, fi and f∞ with a family of hypergeometric functions.展开更多
Applying an addition formula of Liu(2007),we deduce certain Jacobi theta function identities.From these results we confirm several q-trigonometric identities con.jectured by Gosper(2001).Another conjectured identity o...Applying an addition formula of Liu(2007),we deduce certain Jacobi theta function identities.From these results we confirm several q-trigonometric identities con.jectured by Gosper(2001).Another conjectured identity on the constant Πq is also settled.展开更多
文摘Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results and the consequences assuming RH are true.
文摘1 Introduction Jacobi forms are the generalization of Jacobi theta series and the coefficients of the Fourier-Jacobi expansion of a Siegel modular form. The theory develops systematically in recent years and has many interesting applications in theory of modular forms and number theory.
文摘In this review article, we revisit derivation of the cumulative density function (CDF) of the test statistic of the one-sample Kolmogorov-Smirnov test. Even though several such proofs already exist, they often leave out essential details necessary for proper understanding of the individual steps. Our goal is filling in these gaps, to make our presentation accessible to advanced undergraduates. We also propose a simple formula capable of approximating the exact distribution to a sufficient accuracy for any practical sample size.
基金Supported by Innovation Program of Shanghai Municipal Education Commission and PCSIRT
文摘In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.
文摘By Witten rigidity theorem and the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, the elliptic genus of a homogeneous spin manifold G/H can be expressed as a sum of theta functions quotients over the Weyl group of G. Consequently, we obtainseveral classes of combinatorial identities of theta functions.
文摘We shall study the differential equation y^l2=Tn（y）-（1-2μ2）;where μ2 is a constant, Tn（x） are the Chebyshev polynomials with n = 3,4,6. The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on 2F1 （1/4, 3/4; 1; z）, 2F1 （l/3, 2/3; 1; z）, 2F1 （1/6, 5/6; 1; z） and provide a unified approach to a set of identities of Rmanujan involving these hypergeometric functions.
文摘Let G be the group of the fractional linear transformations generated by T（τ）=τ ＋ λ, S（τ）=（τ cos π/n ＋ sin π/n）/（-τ sin π/n ＋ cos π/n）;where λ=2（cos π/m ＋ cos π/n）/sin π/n;m, n is a pair of integers with either n ≥ 2, m ≥ 3 or n ≥ 3, m ≥ 2; τ lies in the upper half plane H.A fundamental set of functions f0, fi and f∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan＇s triple differential equations associated with the group G and establish the connection of f0, fi and f∞ with a family of hypergeometric functions.
基金supported by National Natural Science Foundation of China(Grant No.11801451).supported by National Natural Science Foundation of China(Grant No.11371184)the Natural Science Foundation of Henan Province(Grant Nos.162300410086,2016B259 and 172102410069)。
文摘Applying an addition formula of Liu(2007),we deduce certain Jacobi theta function identities.From these results we confirm several q-trigonometric identities con.jectured by Gosper(2001).Another conjectured identity on the constant Πq is also settled.
