In this paper, a common generalization of the Rogers-Ramanujan series and the generating function for basis partitions is studied. This leads naturally to a sequence of polynomials, called BsP-polynomials. In turn, the BsP-polynomials provide simultaneously a proof of the Rogers-Ramanujan identities and a new, more rapidly converging series expansion for the basis partition generating function....

1. Complex differentiation 2. Exponentials, trigonometric functions 3. Differentiating power series: Abel’s theorem 4. Path integrals 5. Cauchy’s theorem 6. Cauchy’s formula 7. Power series expansions, Morera’s theorem 8. Identity principle 9. Liouville’s theorem: bounded entire functions are constant 10. Laurent expansions around singularities 11. Residues and evaluation of integrals 12. Logar...

This is a survey and expository article. Some new developments on refinements, generalizations, applications of Jordan’s inequality and related problems, including some results about Wilker-Anglesio’s inequality, some estimates for three kinds of complete elliptic integrals and several inequalities for the remainder of power series expansion of ex, are summarized.

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n, s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the multivariate sigma function. This expression is in turn used to prove fundamental properties on the series expansion of the sigma function established in a previous paper in a different method.

What is the dimension of a smooth family of complex Hadamard matrices including the Fourier matrix? We address this problem with a power series expansion. Studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6, or a power of a prime. In general the answer depends critically on the prime number decomposition of the dimension. Our results sug...

In this paper, we tried to evaluate the fractional derivatives by using the Chebyshev series expansion. We discuss the indefinite quadrature rule to estimate the fractional derivatives of Riemann-Liouville type.

The Catalan-Larcombe-French sequence {Pn}n≥0 arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence {P2 n – Pn–1Pn+1}n≥1 are strictly log-con...

The average concentration of a passive scalar advected from a point source by a multivariate normal velocity field is shown to deviate from a Gaussian profile. The flatness (kurtosis) is calculated using an asymptotic series expansion valid for velocity fields with short correlation times or weak space dependence. An explicit formula for the excess flatness at first order demonstrates maximum d...

In this study,differential transform method (DTM) is applied to linear and nonlinear system of ordinary differential equations. If the system considered has a solution in terms of the series expansion of known functions,this powerful method catches the exact solution.So as to show this capability and robustness, some systems of ordinary differential equations are solved as numerical examples.

What is a least integer upper bound on van der Waerden number W (r, k) among the powers of the integer r? We show how this can be found by expanding the integer W (r, k) into powers of r. Doing this enables us to find both a least upper bound and a greatest lower bound on W (r, k) that are some powers of r and where the greatest lower bound is equal to or smaller than W (r, k). A finite series ...