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# HANKEL TYPE INTEGRAL TRANSFORMS CONNECTED WITH THE HYPER-BESSEL DIFFERENTIAL OPERATORS

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INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 2000

## HANKEL TYPE INTEGRAL TRANSFORMS CONNECTEDWITH THE HYPER-BESSEL DIFFERENTIAL OPERATORS

−β

m

j=1

j

j

c

∞ 0

[155]

(2)

s

0

ν

∞ 0

ν

ν

ν

−2

dxd

dxd

ν

−β

m

j=1

j

j

m

dxdmm

ν

−1γ

(3)

−1γ

σ

−s

γ

−1γ

M−1

γ (L)

σ

γ

−1γ

−1

−1γ

λ

12

λ

−1γ

λ

12

γ

L(σ(τ ))

λ

λ

12

γ

λ

−1γ

−1γ

0

s−1

E



it−1/2

−1/2

(4)

−1

∞ 0

σ

−s

∞ 0

−1

∞ 0

β

∞ 0

m,00,2m

β

j

1β

2m1

j

m,np,q

j

p1

j

q1

L

−s

m

j=1

j

n

j=1

j

p

j=n+1

j

q

j=m+1

j

j

j

j

j

−∞

+∞

i∞

(5)

2m

j=1

j

j

j

−1

β

∞ 0

m,00,2m

β

j

2mm+1

j

m1

m,00,2m

j

12

1

2m1

E



β2

β212

β

it−1/2

Eη

β

η

β

itβ12

it

β2

β212

m,00,2m

β

j

12

1

2m1

m,00,2m

β

j

β1

2m1

s−12

m

j=1

j

1β

βs

2m

j=m+1

β1

j

βs

s−12

2m

j=m+1

j

βs

m

j=1

j

sβ

m,00,2m

β

j

2mm+1

j

m1

(6)

β

−1γ

2m

j=1

j

j

β

β

m

β

β

β

−β

2m

2m

j=1

j

β

σ

−s

β

−1γ

β

−β

m

σ

−s

β

m

σ

−s

m

j=1

j

2m

j=m+1

j

p,qm,n

σ

−s

m

j=1

j

j

n

j=1

j

j

p

j=n+1

j

j

q

j=m+1

j

j

j

j

j

j

λ

−1γ

β

β

β

m

σ

−s

m

β

m

(7)

λ

−1γ

β

β

m

σ

−s

m

β

σ

−s

m

β

β

j

j+m

1

β

β

∞ 0

m,00,2m

β

j

β1

m1

j

m1

−1

j

β

−1γ

j

β

β

m

β

β

β

−β

2m

m

j=1

j

m

j=1

j

1

s−12

12

2s

2s

(8)

s

s

2

s

s

−2

2

2

1

12

s−12

s2

12

s2

c

c

2

c

c

−2

2

2

12

1

ν2

14

ν

s−12

14

ν2

s2

34

ν2

s2

ν

2

ν

ν

−2

2

2

1 4

2

2

0

0

0

0

0

1

2

58

0

0

s−12

18

s4

18

s4

38

s4

38

s4

4

(9)

−4

2

2

ν

−2

τ,αδ

n

j=1

δτ +α,n−α

δτ,α

1

Γ(α)

1

0

α−1

τ

1/δ

ν 212,12

2

12

ν2

12

ν+1

2

12

2ν212,12

ν+1

2

−1γ

12

−1γ−1 2

∞ 0

ν

2

−1γ

52

ν

2

ν

1

ν2

2

ν2

s−12

ν2

12

2s

ν2

12

2s

(10)

1

2

1

ν2

s2

ν2

12

s2

2

s−12

ν2

12

s2

ν2

s2

2

12

1

−1γ

1

2

12

ν2

1

2

−1γ

12

−1γ−1

2

s−12

ν2

s2

ν2

s2

(11)

### - Boston - London, 1994.

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