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# On the linear independence of values of \$G\$-functions

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(1)

(2)

k=0

k

k

É0

β[s],n

k=0

k

s

k+n

α,β,S

β[s,n]

K,F,β

F,β

K,F,β

K

α,β,S

F,β

K,F,β

k=0

k

k

1

k

1k+1

k

k

k

2

0

k

2k+1

0

k

0

k

0

k

k=0

k

k

k

k=0

k

k=1

k

s

k=1

k

s

k=0

k

2k+1

1

ν

ν

1

ν−1

É0

ν−1

ν

ν−1

k=0

1

k

ν

k

1

k

ν−1

k

k

(3)

k

0

1

1

µ

Cu

i

u

µ

k=0

k

k

É0

β,n[s]

k=0

k

s

k+n

α,β,S

K

β[s],n

α,β,S

K

α,β,S

0

z

0

β,n[s]

α,β,S

s

s

k=1

k

s

k

0,n[s]

n,s

s

(4)

β[s],n

S,r,n

β,n[s]

ω−µ

`

j=0

j

j

j

K

1

2

1

0

`

0

max(1,`−1)

2

0

3

0

0

0

µ

i=1

i

`

`

µ

i=1

i

i

j

0

0

1

2

1

2

(5)

β[s],n

1

β,n[s]

j

K

µ−ω

`

j=0

j

j

j,β

µ

j

β

`

j=0

j

j,β

β

1

µ

A

i

µ

A

A

k=0

k

k

β

1

µ

C

µ

β

1

µ

C+βIµ

m

β

β

m

k

β

k

β+k

β+k

β

k

β

k

(6)

j

dj

m=0

j,m

m

β

β

µ

`

j=0 dj

m=0

j

j,m

m

β

µ

β

`

j=0 dj

m=0

j

j,m

m

µ

β

`

j=0

j

j

β

β

β

−β

−β

−β

β

β

β

β,n[s]

nÊ1

`

j=0

j,β

β[s],n

+j

`

j=0 s−1

t=1

j,n,t,s,β

β[t],n

+j

`

j=0

n+j

j,n,s,β

j,n,t,s,β

K

j,n,s,β

K

j

z

0

β+u

k=0

k

z

0

β+u+k

k=0

k

β+u+k+1

β

β[1],u

+1

1

µ−ω

z

0

β+n1

1

`

j=0 dj

m=0

j,m

m p=0

mp

z

0

β+n+j1

p

z

0

β+n+j1

p

β+n+j

p−1

q=0

pq1

pq1

q

p

p

β

β,n[1]+j

β

dj

m=0

j,m m

p=0

p

p

m−p

j

`

j=0

j

β[1],n+j

`

j=0

n+j

j,n,1,β

(7)

j,n,1,β

dj−1

q=0

j,n,1,q,β

q

j,n,1,q,β

dj

m=0

j,m m

p=q+1

mp

pq−1

pq−1

µ

µ

j,n,1,β

K

j

z

0

β−1

β,n+j[s]

β

β,n+j[s+1]

`

j=0

j,β

β,n+[s+1]j

β

z 0

`

j=0

j,β

β−1

β,n+[s] j

β

`

j=0 s−1

t=1

j,n,t,s,β

z 0

β−1

β,n+j[t]

β

`

j=0

z 0

β+n+j−1

j,n,s,β

`

j=0 s−1

t=1

j,n,t,s,β

β,n[t+1]

+j

`

j=0 dj−s

q=0

j,n,s,q,β

β

z 0

β+n+j−1

q

`

j=0

j

β,n+j[s+1]

`

j=0 s

t=1

j,n,t,s+1,β

β,n+[t] j

`

j=0

n+j

j,n,s+1,β

j,n,t,s+1,β

j,n,t−1,s,β

` i=0

dj−s

q=0

q

q

i,n,s,q,β

j,n,s+1,β

djs

q=0

j,n,s,q,β

q−1

p=0

qp1

qp1

p

K

j

β,n[s]

n∈N, 0ÉsÉS

j,t,s,n,β

j,s,n,β

β[s],n

s

t=1

`+m−1

j=1

j,t,s,n,β

β[t,j]

µ−1

j=0

j,s,n,β

j

(8)

1

j,t,s,n,β

j,s,n,β

1/n

1

S

j,t,s,n0

0

j,s,n0,β

0

2

1/n

2

S

β

`

j=0

j

j,β

1

2

1

j,t,s,n,β

j,s,n,β

0

0

k

k

0

k

k→+∞

k

0

k

S,r,n

β,u[s]

S,r,n

S−r

k=0

r n

Sn+1

k

−k

1

S,r,n

α,β,S

S,r,n

β,u[s]

S,r,n

(9)

0

u,s,n

u,n

S,r,n

`0(β)

u=1

S s=1

u,s,n

β,u[s]

µ−1

u=0

u,n

S(`−1)

u

n

S−r

S

S

n+1

j=1 S

s=1

j,s,n

s

j,s,n

S,r,n

n+1

j=1 S

s=1

j,s,n

j

β,j[s]

S,r,n

`0(β)

u=1

S s=1

u,s,n

β,u[s]

µ−1

u=0

u,n

S(`−1)

u

u,s,n

u,s,n

u

n+1

j=`0(β)+1 S

σ=s

j

j,σ,n

u,s,σ,j

u,n

n+1

j=`0(β)+1 S

s=1

j,s,n

j+S(`−1)

u,s,j

u,s,n

u,n

n→+∞

u,s

u,s,n

1/n

1

S

r

S+r+1

n→+∞

u,s

u,s,n

1/n

1

S

r

S+r+1

0

0

j0,s0,n

|z+β+j0|=1/2

n

0

s0−1

(10)

n

0

r n

r n−1

k=0

r n−1

k=0

0

0

r n−1

k=0

0

r n−1

k=0

0

0

r n

0

r n

0

0

n+1

n

k=0

n

k=0

0

0

n k=0

0

j0−3

k=0

0

3

n k=j0+1

0

0

0

n

S−r

0

0

0

S

0

S

S

0

0

0

S

0

S

S

S−r

0

0

r

0

S

S

S

S

j0,s,n

β+e j0+r n+1

r n

S(n−2)

S

S

S

β+(r+1)n+1e

r n

S(n−2)

S

S

S

0

n→+∞

jÉn+1

j,s,n

1/n

r

S+r+1

u,s,n

u,n

K

n

nÊ1

n

u,s,n

K

n

u,n

K

n→+∞

1/nn

2

S

2r

S

n

n

Sr

r n

Sn+1

(11)

n

0

j0+1,s,n

S−s

n

0

S

|t=−β−j0

λ

λ

λ

n

0

S

r

`=1

`

Sr

`

n

n+1

0

n+1

0

`

n

p=0 p6=j0

0

p,`,n

p,`,n

np

n

p=0 p6=j0

0

p,n

p,n

p

p,n

λ

`

|t=−β−j0

0,λ

n

p=0 p6=j0

λ

0

p,`,n

0

λ+1

0,λ

n

p=0 p6=j0

p,`,n

0

λ

0,λ

λ

|t=−β−j0

n

p=0

p,n

0

λ

nλ

(1)n

λ

`

|t=−β−j0

nλ

λ

|t=−β−j0

n

(1)n

p,`,n

(1)n

2n

Ss

n

0

S

µ

µ1

1

µr

r

µr+1

µS

1

S

S

1

S

nSs

2r n

j+1,s,n

n

n+o(n)

(12)

S,r,n

1

p

p

j=1

j

k

u

0

p

i

j

j

j

sj

bj

t nj

j

j

j

1

p

S,r,n,j

c+i

ci

j

Sr

S

S

t n

S,r,n

p

j=1

r n

S,r,n,j

S,r,n,j

S,r,n,j

S,r,n,j

(Sr+2)/2

j

sj

(S+r)/2+bj

c+i∞

ci

j,β

nϕ(−α/ξj,t)

j,β

(S+1)/2+Sβ−bj

1/2

S(2β+3)/2

j,β

j,β

S,r

0

0

S/(r+1)

(13)

S,r,n,j

(S−r+3)/2

j

j,β

j

sj

ϕjn

(S+r+1)/2j

j

S,r

j

j

j

j

j

00

j

j

j,β

j,β

j

j

j,β

j

j

j

j

2

j

(S+1)/2+Sβ−bj

1/2

S(2β+3)/2

S,r

j,β

(S+1)/2+Sβ

bj

1/2

S(2β+3)/2

S+1

S

S+1

S

j,β

Sβ−(S+1)/2

bj

1/2

S(β+1)

ϕj

ω

−S/(r+1)

2

2

1,β

Q,β

1

Q

q

S,r,n

n

κ

λ

Q

q=1

q,β

nq

S−r

S,r,n

p

j=1

r n

S,r,n,j

S,r,n,j

S,r,n

r n

(S−r+3)/2

(S+r1)/2

p

j=1

j

j

j

−βj

sj

ϕjn

(14)

1

Q

j

j

j

3

S,r,n

n

κ

s

Q

q=1

q,β

nq

q

jq

q,β

r n

(S−r+3)/2

jq

jq

jq

q

j

ω

−S/(r+1)

α,β,S

K

u,s,n

n

u,s,n

u,n

n

u,n

K

n→+∞

u,s

u,s,n1/n

u,n

1/n

1

S

2

S

2r

S

r

S+r+1

n

n

S,r,n

`0(β)

u=1 S

s=1

u,s,n

β[s],u

µ−1

u=0

u,n

S(`−1)

u

2

n

n(10 +o(1))

Q

q=1

q,β

nq

0

2

S

2r

S

Sr

α,β,S

β[s],u

v

0

K

α,β,S

0

1

2

0

1

2

### 2q max(1, 1/ α ) ´

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