A new method q-calculus

Abstract: In q-calculus we are looking for q-analogues of mathematical objects, which have the original object as limits when q tends to 1. q-Calculus has wide-ranging applications in analytic number theory and theoretical physics. The main topic of the thesis is the invention of the tilde operator and the renaissance of the q-addition. There are two types of q-addition, the Ward-AlSalam q-addition and the Hahn q-addition. The first is both commutative and associative, while the second is neither. This is one of the reasons why sometimes more than one q-analogue exist. These two operators form the basis of the method which unites hypergeometric series and q-hypergeometric series and which gives many formulas of q-calculus a natural form reminding directly of their classical origin. This method is reminiscent of Heine, who mentioned the case where one parameter in a q-hypergeometric series is plus infinity. The q-addition is the natural way to extend addition to the q-case as is shown when restating addition formulas for q-trigonometric functions.We give a more lucid definition of the q-difference operator. A new notation for powers of q reminding of the exponential function is given. A q-Taylor formula with remainder term expressed as q-integral is proved.We present a new expression for generalized Vandermonde determinants, and thus for the Schur function. We also obtain an equivalence relation on the set of all generalized Vandermonde determinants. We find a more general expression for the Vandermonde determinant. We show the connection to a determinant of Flowe and Harris and to the solution of difference and q-difference equations with constant coefficients.