Hello. I need help with this question on Advanced Math topic. Thank you.
Prove the following recurrence equations related to Bessel functions
Expert Answer
We first consider the definition of Bessel function of order n and differentiate it w.r.t. x.
Let us now multiply both sides by x, in the above derivative of Jn. After this we add the powers of two (x/2) terms. This will help us change the power of (x/2) back to its original value.
Next, we split the summation in to two parts by opening the brackets of the term (n + 2r). By doing this, we observe that the first summation is the definition of Jn. In the second summation, we expand it for r = 0 and have new summation starting from r = 1. Once we do that, r can be canceled out from numerator and denominator.
To simplify the second summation further, we substitute, (r – 1) = s that is r = s + 1. This will change the variable of summation to s and the new lower limit for s will be r – 1 = 1 – 1 = 0. After taking a minus sign out and rearranging few terms, we observe that the second summation becomes the definition of Jn+1.
To prove the second recursion, we start with xJn’ equation we got in step 2. In this equation we replace n with –n + 2n. We then split the summation by opening the brackets of the term (–n + 2n + 2r).
We then cancel out (n + r) term from numerator and denominator. After rearranging the terms, we observe that the second summation becomes the definition of Jn-1.
