Phytoplankton are tiny floating plants (algae) living in oceans. In the process of photosynthesis, phytoplankton produces half of the world's oxygen. Moreover, by primary production, death and sinking, they transport carbon from the ocean's surface layer to marine sediments. There are many species of phytoplankton that can be distinguished according to morphology. In this paper, we investigate ...

In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon e...

We give some new q-supercongruences on truncated forms of squares basic hypergeometric series. Most them are modulo the cube a cyclotomic polynomial, and two fourth power polynomial. The main ingredients our proofs creative microscoping method, lemma El Bachraoui, Chinese remainder theorem for coprime polynomials. also propose several related conjectures further study.

A possibly unknown approach to the problem of finding the common term of a power series is considered. A direct formula for evaluating this common term has been obtained. This formula provides useful expressions for direct evaluation of the number of partitions of a nonnegative integer and the partitions themselves. These expressions permit easily to work with power series, evaluate nth derivat...

In this paper, we derive a simple and efficient matrix formulation using Laguerre polynomials to solve the singular integral equation with degenerate kernel. This method is based on replacement of unknown function by truncated series well known expansion functions. leads system algebraic equations coefficients. Thus, solving equation, coefficients are obtained. Some numerical examples included ...

In 2017, He (2017) [9] established two supercongruences on truncated hypergeometric series and further proposed related conjectures. Subsequently, Liu [11] extended He's formulas confirmed the second conjecture. However, first conjecture is still open up to now. With help of creative microscoping method Chinese remainder theorem for coprime polynomials, we derive several q-supercongruences modu...

formulae expressing explicitly the coefficients of an expansion of double jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. extension to expansion of triple jacobi polynomials is given. the results for the special cases of double and triple ultraspher...