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...

-Now a day’s security is very essential measure during the data transmission ,in this paper we are proposing an efficient mechanism of group key protocol and high complex cryptographic algorithm. we consider how the chaotic maps can be improved to fulfill the cryptographic demands. To achieve this, we extend the domain of Chebyshev maps from real number set to finite field. Based on this extens...

In this paper, we consider the doubly indexed sequence a(r) ? (n,m), (n,m 0), defined by a recurrence relation and an initial (0,m), (m 0). We derive with help of some differential operator explicit expression for (n, in term degenerate r-Stirling numbers second kind sequence. observe that 0) = ?n,?(r), (0,m) 1/m+1 , En,?(r), (1/2)m . Here ?n,?(x) En,?(x) are fully Bernoulli polynomials Euler p...

We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.

We've been able to show recently that Permutable Chebyshev polynomials (T polynomials) defined over the field of real numbers can be used to create a Diffie-Hellman-like key exchange algorithm and certificates. The cryptosystem was theoretically proven to withstand attacks using quantum computers. We additionally prove that attacks based on the inverse of the cosine representation of T polynomi...

The resultant is an algebraic expression, computable in a finite number of arithmetic operations from the coefficients of two univariate polynomials, that vanishes if, and only if, the two polynomials have common zeros. The paper considers formal resultant for degree-deficient polynomials (polynomials whose actual degree is lower than their assumed degree). Some key properties of the resultant ...

Many practical problems can be formulated as convex optimization problems over the cone of nonnegative univariate polynomials. We use a cutting plane method for solving this type of optimization problems in primal form. Therefore, we must be able to verify whether a polynomial is nonnegative, i.e. if it does not have real roots or all real roots are multiple of even order. In this paper an effi...