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...

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...

An infinite dimensional algebra which is a non-decomposable reducible representation of su(2) is given. This algebra is defined with respect to two real parameters. If one of these parameters is zero the algebra is the commutative algebra of functions on the sphere, otherwise it is a noncommutative analogue. This is an extension of the algebra normally refered to as the (Berezin) quantum sphere...

The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...

In this paper, we introduce advanced architectures of genetically-oriented Fuzzy Neural Networks (FNNs) based on fuzzy set and fuzzy relation and discuss a comprehensive design methodology. The proposed FNNs are based on ‘if-then’ rule-based networks with the extended structure of the premise and the consequence parts of the fuzzy rules. We consider two types of the FNNs topologies, called here...

A method for response integration in modular neural networks with type-2 fuzzy logic for biometric systems p. 5 Evolving type-2 fuzzy logic controllers for autonomous mobile robots p. 16 Adaptive type-2 fuzzy logic for intelligent home environment p. 26 Interval type-1 non-singleton type-2 TSK fuzzy logic systems using the hybrid training method RLS-BP p. 36 An efficient computational method to...

In this work we give an alterative proof of one of basic properties of zonal polynomials and generalised it for Jack polynomials

In this paper, a recurrent functional neural fuzzy network (RFNFN) with modified differential evolution (MDE) method is proposed to solve the prediction problems. The proposed RFNFN model adopts a functional link neural network (FLNN) to the consequent part of the fuzzy rules. FLNN uses orthogonal polynomials and linearly independent functions to form a functional expansion. Thus, the consequen...

The problem of simultaneous approximations of a given function and its derivatives, has been addressed frequently in pure and applied mathematics. In pure mathematics, Bernstein polynomials get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of a...