Can symsum Create a Matrix of Functions Using a Vector of Summation Limits?

Right now I've been using a for loop to write out functions because I can't figure out how to write them all using the same symsum command:

zi = 2.5;
syms k q
n=1:18;
f(q) = factorial(q)*zi.^(n+1) 
B(k) = symsum(f(q)*(k*zi).^(n-q-0.5).*besselj(n-q+0.5,k*zi),q,0,n)

I want a vector of B(k) functions, one for each value of n, with the summation limit determined by n. I have a bunch of these functions to write, so doing them in a for 1:n loop is extremely slow. Any help will be greatly appreciated!

Answer 1

[Walter Roberson]

symsum() just might be vectorized over its first argument, what to sum, but I would be fairly sure it is not vectorized over a list of upper bounds.

I ran the expression through a number of potential transformations, and found no useful transforms for this situation.

Interestingly, up to n=9, the results look like they are building up a useful pattern, but from n=10 onward, the analytic results come out in terms of besselj() calls that are not easily expressed as simple constants; and as n increases, more and more besselj() terms get added in to the mess.

I think you are going to end up looping just the way you do now.

[Addendum]

When I tried transforms on the result for specific n, I had some success in Maple, first with convert/elementary and then much better, with convert/Elliptic_related . The elliptic analysis produced the same structure of result for each n, with different coefficients. Potentially with sufficient analysis the expression for the coefficients could be derived.

Up to n=18, the results were:

[-(1/4)*2^(1/2)*(25*cos(25/2)-4*sin(25/2))/Pi^(1/2),
 -(5/16)*2^(1/2)*(601*sin(25/2)+200*cos(25/2))/Pi^(1/2),
 (25/64)*2^(1/2)*(13625*cos(25/2)-8542*sin(25/2))/Pi^(1/2),
 (125/256)*2^(1/2)*(260621*sin(25/2)+317350*cos(25/2))/Pi^(1/2),
 -(625/1024)*2^(1/2)*(2901925*cos(25/2)-10124744*sin(25/2))/Pi^(1/2),
 -(3125/4096)*2^(1/2)*(-58400203*sin(25/2)[phone]*cos(25/2))/Pi^(1/2),
 -(78125/16384)*2^(1/2)*(1015996175*cos(25/2)[phone]*sin(25/2))/Pi^(1/2),
 (32421875/65536)*2^(1/2)*(-444369281*sin(25/2)[phone]*cos(25/2))/Pi^(1/2),
 (1953125/262144)*2^(1/2)*(750818666075*cos(25/2)-635481183124*sin(25/2))/Pi^(1/2),
 (9765625/1048576)*2^(1/2)*(26639137973200*cos(25/2)-769031287741*sin(25/2))/Pi^(1/2),
 (48828125/4194304)*2^(1/2)*(644049903002525*cos(25/2)[phone]*sin(25/2))/Pi^(1/2),
 (8544921875/16777216)*2^(1/2)*(363812095059250*cos(25/2)[phone]*sin(25/2))/Pi^(1/2),
 (152587890625/67108864)*2^(1/2)*(1819806922548547*cos(25/2)+5178280343225296*sin(25/2))/Pi^(1/2),
 (30517578125/268435456)*2^(1/2)*(835854617746266100*cos(25/2)+4754359151282421419*sin(25/2))/Pi^(1/2),
 (152587890625/1073741824)*2^(1/2)*(19476608985999283925*cos(25/2)+193620992439805397246*sin(25/2))/Pi^(1/2),
 (762939453125/4294967296)*2^(1/2)*(674265625341945660850*cos(25/2)+8987193058841489688977*sin(25/2))/Pi^(1/2),
 (19073486328125/17179869184)*2^(1/2)*(6417929201021471562475*cos(25/2)+94058741322934234202548*sin(25/2))/Pi^(1/2),
 (95367431640625/68719476736)*2^(1/2)*(363528164903755375039000*cos(25/2)+5443917848856746781665411*sin(25/2))/Pi^(1/2)]