Why is a symbol not being assigned in a function

Question

The function should be evaluating at j=1 but I am getting j in the final output as a variable

I have the following function:

g [j_, k_] := Sum[Sum[Binomial[j, μ]  Binomial[k, ν] ((1/2)^(j + k) (-1)^(ν + Floor[1/2 k])  B[k])/((β τ + 1)^2 + φ^2 (j + k - 2 μ - 2 ν)^2), {ν, 0,k}], {μ, 0, j}];

where, B[k] is defined by:

B[k_] := If[ IntegerQ[k/2], β τ + 1, (j + k - 2 μ - 2 ν) φ]; 

However, if I try and evaluate the following expression:

gp[j_, k_] := ρ  g[ j, k];
gp[1,1]

I get a j which clearly should have been set to 1

I've tried various combinations of Evaluate[] and := assignments but none seem to be working. I can get temporary correct behaviour by doing B[j_, k_] := Evaluate[...] and calling B[j,k] from g[j_, k_] but this then reverts to a failed state when I save it and call as a packaged function

Expected Output

The expected output is:

(ρ φ)/((1 + β τ)^2 + 4 φ^2)

Instead I get:

ρ (-(((-3 + j) φ)/(4 ((1 + β τ)^2 + 4 φ^2))) + ((1 + j) φ)/(4 ((1 + β τ)^2 + 4 φ^2)))

Answer 1

The issue you are experiencing is that you define j as a local variable to the functions gp and g but using it as a global variable in your function B. So in effect it is undefined for use with B.

Defining j local to b will resolve your issue. I also changed B[k_] into lowercase, so your new function header is b[j_,k_] (also in your function g) as per Mathematica conventions.

b[j_, k_] := 
  If[IntegerQ[k/2], β τ + 1, (j + k - 2 μ - 2ν)/ φ];

g[j_, k_] := 
 Sum[
  Sum[
   Binomial[j,μ] Binomial[k,ν] ((1/2)^(j + k) (-1)^(ν + Floor[1/2 k]) b[j,k])/((β τ + 1)^2 
   + φ^2 (j + k - 2 μ - 2 ν)^2)
  , {ν, 0, k}]
 , {μ, 0, j}];

Returns what you are expecting:

(ρ φ)/((1 + β τ)^2 + 4 φ^2)

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As @ciao stated; it is recommended to not use uppercase in function definitions, and especially not for symbols. This is the reason for using b instead of B. Although in your case using an uppercase symbol does not cause any problems, check the result of ?D as an example why it is not a good idea to use uppercase initials).