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Full-Text Articles in Physical Sciences and Mathematics
The Q-Gauss Product, Q-Trigonometry Via Landen-Like Identities, And Positive Alternating Q-Series, Sarah Abo Touk
The Q-Gauss Product, Q-Trigonometry Via Landen-Like Identities, And Positive Alternating Q-Series, Sarah Abo Touk
Theses
The object of this report is q-series and their relationship with certain special functions. Firstly, Jackson’s q-analogue of the Euler gamma function is introduced and a q-analogue for a famous formula of Gauss on products of the gamma function will be presented. Secondly, Jacobi’s theta functions will be discussed in details and new Landenlike half argument formulas will be established. As an application, q-trigonometric formulas shall be derived and a new proof for a well-known q-series relation of Jacobi will be given. Thirdly, an extended Bailey transform will be presented and a variety of new q-series will be deduced as …
Sum Of Squares With Q-Series, Gosper’S Q- Trigonometry, An New Identities Via An Extended Bailey Transform, Zina Al Houchan
Sum Of Squares With Q-Series, Gosper’S Q- Trigonometry, An New Identities Via An Extended Bailey Transform, Zina Al Houchan
Theses
This report is concerned about q-series and some of their applications. Firstly, Jacobi’s q-series proof for Legendre’s theorem on sums of four squares will be presented. By way of comparison, the classical approach of this result will be also discussed. Secondly, Gosper’s q-trigonometry will be introduced using Jacobi’s theta functions and the theory of elliptic functions shall be employed to confirm one of Gosper’s conjectures. As an application, a proof for Fermat’s theorem on the sums of squares will be provided. Thirdly, an extended version of Bailey’s transform will be established and as a consequence, a variety of new q-series …