Hi,
Btw, the series computed by sympy seem to be correct.
Here is the comparison to Axiom/Fricas:
(13) -> l := 0
(13) 0
Type: NonNegativeInteger
(14) -> series((2*l+1)*integrate(exp(-r*a)*legendreP(l, (-r^2 + t^2 +
1)/(2*t)), r=(1-t)..(1+t))/(2*t) / exp(-a), a=0)
2 4 6 8 10
t 2 t 4 t 6 t 8 t 10 11
(14) 1 + -- a + --- a + ---- a + ------ a + -------- a + O(a )
6 120 5040 362880 39916800
Type: UnivariatePuiseuxSeries(Expression(Integer),a,0)
(15) -> l := 1
(15) 1
Type: PositiveInteger
(16) -> series((2*l+1)*integrate(exp(-r*a)*legendreP(l, (-r^2 + t^2 +
1)/(2*t)), r=(1-t)..(1+t))/(2*t) / exp(-a), a=0)
(16)
3 3 5 5 7 7
t 2 t 3 t 4 t 5 t 6 t 7
t + t a + -- a + -- a + --- a + --- a + ----- a + ----- a
10 10 280 280 15120 15120
+
9 9 11
t 8 t 9 t 10 11
------- a + ------- a + --------- a + O(a )
1330560 1330560 172972800
Type: UnivariatePuiseuxSeries(Expression(Integer),a,0)
(17) -> l := 2
(17) 2
Type: PositiveInteger
(18) -> series((2*l+1)*integrate(exp(-r*a)*legendreP(l, (-r^2 + t^2 +
1)/(2*t)), r=(1-t)..(1+t))/(2*t) / exp(-a), a=0)
(18)
4 2 4 6 4 6 8 6
2 2 3t + 14t 2 t 3 t + 12t 4 t 5 t + 22t 6
t + t a + ---------- a + -- a + --------- a + --- a + --------- a
42 14 504 504 33264
+
8 10 8 10 12 10
t 7 3t + 104t 8 t 9 t + 50t 10 11
----- a + ------------ a + ------- a + ----------- a + O(a )
33264 10378368 3459456 518918400
Type: UnivariatePuiseuxSeries(Expression(Integer),a,0)
(19) -> l := 3
(19) 3
Type: PositiveInteger
(20) -> series((2*l+1)*integrate(exp(-r*a)*legendreP(l, (-r^2 + t^2 +
1)/(2*t)), r=(1-t)..(1+t))/(2*t) / exp(-a), a=0)
(20)
5 3 5 3 7 5 7 5
3 3 5t + 36t 2 5t + 6t 3 5t + 88t 4 15t + 44t 5
t + t a + ---------- a + --------- a + ---------- a + ----------- a
90 90 3960 11880
+
9 7 9 7 11 9 11 9
5t + 156t 6 5t + 26t 7 t + 48t 8 t + 8t 9
----------- a + ---------- a + ---------- a + --------- a
308880 308880 7413120 7413120
+
13 11
t + 68t 10 11
----------- a + O(a )
1260230400
Type: UnivariatePuiseuxSeries(Expression(Integer),a,0)
(21) -> l := 4
(21) 4
Type: PositiveInteger
(22) -> series((2*l+1)*integrate(exp(-r*a)*legendreP(l, (-r^2 + t^2 +
1)/(2*t)), r=(1-t)..(1+t))/(2*t) / exp(-a), a=0)
(22)
6 4 6 4 8 6 4
4 4 7t + 66t 2 21t + 44t 3 105t + 2340t + 1144t 4
t + t a + ---------- a + ----------- a + ----------------------- a
154 462 120120
+
8 6 10 8 6 10 8
21t + 104t 5 7t + 270t + 312t 6 7t + 60t 7
------------ a + -------------------- a + ----------- a
24024 720720 720720
+
12 10 8 12 10
7t + 408t + 816t 8 21t + 272t 9
--------------------- a + -------------- a
98017920 294053760
+
14 12 10
21t + 1710t + 5168t 10 11
------------------------- a + O(a )
55870214400
Type: UnivariatePuiseuxSeries(Expression(Integer),a,0)
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