Q-Series with Applications to Binomial Coefficients Integer Partitions, And Sums of Squares
In this report we shall introduce q- Series and we shall discus some of their application to the integer partition , the sums of squares , and the binomial coefficient . We will present the basic theory of q- series including the most famous theorem and rules governing the e object such as the q-binomial theorem and the Jacobi's triple identity. We hall present the q-binomial coefficients which roughly speaking connect the binomial coefficient to q- Series , we will give the most important results on q-binomial coefficients,and we shall provide some of our new result on the divisibility of binomial coefficients. Moreover, we hall give some well-known applications of q-series to sums of two squares and to integer partition such as Ramanujan's modulo 5 congruence.