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Then collecting the results we finally get:

Lemma 1:Let fi(x) be a sequence of functions
indexed by i. Then one can interchange integra-
tion and summation according to the following
rule:

.

We prove this lemma by dividing the axis into
pieces between integers, and interchanging inte-
gration and summation (and collecting) as done
below.

x= 0

t

∫ fk

k= 0

⎣⎦x

∑ (x)dx=

i= 0

⎣⎦t−^1

∑

x=i

i+ 1

∫ fk

k= 0

i

∑ (x)dx+

x=⎣⎦t

t

∫ fk

k= 0

⎣⎦t

∑ (x)dx=

i= 0

⎣⎦t−^1

∑

k= 0

i

∑ fk

x=i

i+ 1

∫ (x)dx+

k= 0

⎣⎦t

∑

x=⎣⎦t

t

∫ fk(x)dx=

k= 0

⎣⎦t−^1

∑

i=k

⎣⎦t−^1

∑ fk

x=i

i+ 1

∫ (x)dx+

k= 0

⎣⎦t

∑

x=⎣⎦t

t

∫ fk(x)dx=

k= 0

⎣⎦t−^1

∑

i=k

⎣⎦t−^1

∑

x=i

i+ 1

∫ fk(x)dx+

k= 0

⎣⎦t−^1

∑

x=⎣⎦t

t

∫ fk(x)dx+

x=⎣⎦t

t

∫ f⎣⎦t(x)dx=

k= 0

⎣⎦t−^1

∑

x=k

t

∫ fk(x)dx+

x=⎣⎦t

t

∫ f⎣⎦t(x)dx=

k= 0

⎣⎦t

∑

x=k

t

∫ fk(x)dx

x= 0

t

∫ fk

k= 0

⎣⎦x

∑ (x)dx=

k= 0

⎣⎦t

∑ fk

x=k

t

∫ (x)dx

Ft(;μ,ρ)= μ

μ+ρk= 0

⎣⎦t

∑

ρ

ρ+μ

⎛

⎝

⎜

⎞

⎠

⎟

k

(qt(−k;ρ)−(^1 −ρ)eμ()k−t).