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I use the code

fv = RecurrenceTable[{a[i] == 
     1/(i (i + 1)) (1/Sqrt[w] a[i - 1] + (2 i - 1 - e[i]) a[i - 2]), 
    a[2] == 1/6 (1/(2*w)+3 - e[i]), a[1] == 1/(2 Sqrt[w])}, a, {i, 3, 10}];

to evaluate the three-step recursion relation where 

e[i_]:=e[i]=2i+1

after that I need to obtain the roots of fv, so I use 

sol2 = N[Solve[fv[[1]] == 0, w]]

my main problem is that e must be calculated using max i, namely the code must calculate the value of e first, then must use i to calculate recursion relation, For example for i=3 , e becomes 7, then a[3] must be calculated that includes e[3] and a[2] and a1, where a[2] must be calculated using e[3] again not e[2], maybe some sort of a loop needed. I really don't know how to address this problem. Any idea?

The correct results for what I intend are:
i=3, w=0.05;
i=4, w=0.0182 and w=0.1900;
and so on.

You can see the original paper which has introduced the above formula and the results here. I have marked important formula. The results have been shown in the table.

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  • $\begingroup$ I'm not entirely sure I understand the question - are you asking how to change your code such that a[i] is computed using e[i] everywhere, even for a[i-1] etc.? $\endgroup$
    – Lukas Lang
    Oct 13 '19 at 17:28
  • $\begingroup$ @LukasLang yes, I explained it in my example. $\endgroup$
    – Wisdom
    Oct 13 '19 at 18:16
  • $\begingroup$ @bbgodfrey sorry it was a mistyping, I corrected it $\endgroup$
    – Wisdom
    Oct 13 '19 at 19:02
  • $\begingroup$ Thanks a lot for your answer. Sorry I forgot to type a 3 in a[2] formula. I corrected it, however just first case (i=3) gives correct answer but other results are wrong, I checked my formula but there was no mistake. Do you think two different roots for i>3 cause the mistake? $\endgroup$
    – Wisdom
    Oct 13 '19 at 20:12
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    $\begingroup$ @Wisdom How did you obtain the values of w in the question for i = 4? Please show the equation you solved. $\endgroup$
    – bbgodfrey
    Oct 13 '19 at 22:00
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Here is an example that doesn't use RecurrenceTable but rather simply defines recursive functions for this (note the addition of a second argument to a):

e[i_] := 2i+1
a[i_] := a[i, e[i]];

a[i_, e_] := 1/(i (i + 1)) (1/Sqrt[w] a[i - 1, e] + (2 i - 1 - e) a[i - 2, e]);
a[2, e_] := 1/6 (1/(2*w) + 3 - e);
a[1, e_] := 1/(2 Sqrt[w])

To create a similar list as you would get from RecurrenceTable:

fv = Table[a[i], {i, 3, 10}]
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  • $\begingroup$ your code does't work and yields a e[3], e[4] and so on. $\endgroup$
    – Wisdom
    Oct 14 '19 at 12:22
  • $\begingroup$ @Wisdom edited to include your definition of e as well (without the memoisation) $\endgroup$
    – Gerli
    Oct 14 '19 at 12:23
  • $\begingroup$ Ohhhh Thank you soooo much, it returned my desire results exactly. $\endgroup$
    – Wisdom
    Oct 14 '19 at 12:30
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With the clarification provided by the reference now given in the question, the desired solution is easily obtained.

a[1] := 1/(2 Sqrt[w]); 
a[2] := 1/6 (1/(2 w) + 3 - e); 
a[i_] := 1/(i (i + 1)) (1/Sqrt[w] a[i - 1] + (2 i - 1 - e) a[i - 2]);
Flatten[Table[NSolve[(a[i] /. e -> 2 i + 1) == 0, w], {i, 3, 10}], {{3}, {1}}][[1]] // Values

(* {{0.05}, 
    {0.190065, 0.0182686}, 
    {0.0404842, 0.0086731}, 
    {0.156634, 0.015429, 0.00478921}, 
    {0.0344873, 0.00753932, 0.00292085}, 
    {0.134788, 0.0134812, 0.00424987, 0.00191167}, 
    {0.0302949, 0.00671401, 0.0026321, 0.00131905},
    {0.119177, 0.012046, 0.00383985, 0.0017433, 0.000948279}} *)

The reciprocals of these values are precisely those given in the reference. Note that Flatten[. . ., {{3}, {1}}][[1]] merely eliminates unnecessary pairs of curly brackets.

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