Mathematics

Let ∑ d|n d≡1(2) 1 d denote the sum of inverses of odd divisors of a positive integer n , and let c r (n) be the number of representations of n as a sum of r squares where representations with different orders and different signs are counted as distinct. The aim is of this note is to prove the following interesting combinatorial identity: ∑ d|n d≡1(2) 1 d = 1 2 ∑ r=1 n (−1 ) n+r r ( n r ) c r (n).

Let B n denote the Bernoulli numbers, and S(n,k) denote the Stirling numbers of the second kind. We prove the following identity B m+n = ∑ 0≤k≤n 0≤l≤m (−1 ) k+l k!l!S(n,k)S(m,l) (k+l+1)( k+l l ) . To the best of our knowledge, the identity is new.

Let ∑ d|n denote sum over divisors of a positive integer n , and t r (n) denote the number of representations of n as a sum of r triangular numbers. Then we prove that ∑ d|n 1+2(−1 ) d d = ∑ r=1 n (−1 ) r r ( n r ) t r (n) using a result of Ono, Robbins and Wahl.

In this manuscript we provide a new polynomial pattern. This pattern allows to find a polynomial expansion of the form x 2m+1 = ??k=1 x ??r=0 m A m,r k r (x?�k ) r , where x,m?�N and A m,r is real coefficient.

We showed that the prime gap for a prime number p is less than or equal to the prime count of the prime number.

We introduce a kind of "perturbation" for the Li-Keiper coefficients around the Koebe function (the K function) and establish a closed system of Equations for the Li-Keiper coefficients. We then check the correctness of some of the many possible solutions offered by the system ,related to the discrete derivative of order n of a function. We also report numerical finding which support our stability conjecture that the tiny part lambda-tiny(n) (the fluctuations around the trend) are bounded in absolute values by gammaxn, where gamma is the Euler-Mascheroni constant.

We introduce methods for deriving analytic solutions from differential-algebraic systems of equations (DAEs), as well as methods for deriving governing equations for analytic characterization which is currently limited to very small systems as it is carried out by hand. Analytic solutions to the system and analytic characterization through governing equations provide insights into the behaviors of DAEs as well as the parametric regions of operation for each potential behavior. For each system (DAEs), and choice of dependent variable, there is a corresponding governing equation which is univariate ODE or PDE that is typically higher order than the constitutive equations of the system. We first introduce a direct formulation for representing systems of linear DAEs. Unlike state space formulations, our formulation follows very directly from the system of constitutive equations without the need for introducing state variables or singular matrices. Using this formulation for the system of constitutive equations (DAEs), we develop methods for deriving analytic expressions for the full solution (complementary and particular) for all dependent variables of systems that consist of constant coefficient ordinary-DAEs and special cases of partial-DAEs. We also develop methods for deriving the governing equation for a chosen dependent variable for the constant coefficient ordinary-DAEs and partial-DAEs as well as special cases of variable coefficient DAEs. The methods can be automated with symbolic coding environments thereby allowing for dealing with systems of any size while retaining analytic nature. This is relevant for interpretable modeling, analytic characterization and estimation, and engineering design in which the objective is to tune parameter values to achieve specific behavior. Such insights cannot directly be obtained using numerical simulations.

The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform. For the first method, two alternative formulae are proposed when adopting the poles inside or outside of the contour, respectively. For the second method, a table on the transform pairs of those popular functions is carefully established. Besides illustrating the effectiveness of the developed methods with two illustrative examples, the applicability are further discussed in the fractional order case.

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, ∫ x α e η x β dx,∫ x α cosh(η x β )dx,∫ x α sinh(η x β )dx,∫ x α cos(η x β )dx and ∫ x α sin(η x β )dx where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1 F 1 and the hypergeometric function 1 F 2 . The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1 F 1 and 1 F 2 . Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and normal distributions are also obtained. The obtained generalized distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square ( χ 2 ) statistical tests and those based on central limit theorem (CLT)).

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function |M(x)| that |M(x)|∼[ 1 π ε √ (x+ε) ] x − − √ (where x is an appropriately large real number, and ε ( 0<ε<1 ) is a small real number which makes 2x+ε to be an integer). For any large x , we can always find an ε , so that |M(x)|<[ 1 π ε √ (x+ε) ] x − − √ .