206 Shy-Der Lin - Shih-Tong Tu - H. M. Srivastava
which, for z 7−→ z 1 − z, assumes the form:
∞
X
k=0
k
n
k M
k
α ; β, z + 1 − z x z
k
= 1 − z
−β
1 − z + z
x
−α n
X
k=0
S n, k M
k
α ; β, x
z 1 − z
k
n ∈ N
; |z| min {1, |x 1 − x|} .
2.5. Ces`aro Polynomials
For the Ces`aro polynomials G
s n
x defined by cf. [14], p. 449, Problem 20 G
s n
x : =
P
n k=0
s+n−k n−k
x
k
=
s+n n
2
F
1
−n, 1; −s − n; x 17
= P
s+1,−s−n−1 n
2x − 1 ,
it is known that [14], p. 449, Problem 20 iii
∞
X
k=0
n + k k
G
s n+k
x t
k
= 1 − t
−s−n−1
1 − xt
−1
G
s n
x 1 − t 1 − xt
n ∈ N ; |t| min
n 1,
|x|
−1
o .
By applying Theorem 1 or Theorem 3, we immediately obtain the following pre- sumably new generating function for the Ces`aro polynomials defined by 17:
∞
X
k=0
k
n
G
s k
x 1 + z 1 + x z
z 1 + z
k
= 1 + z
s+1
1 + x z
n
X
k=0
k S n, k G
s k
x z
k
n ∈ N
; |z| 1 , which, for z 7−→ z 1 − z and x 7−→ x 1 − z 1 − x z, assumes the form:
∞
X
k=0
k
n
G
s k
x z
k
= 1 − z
−s−1
1 − x z
−1
·
n
X
k=0
k S n, k G
s k
x 1 − z 1 − x z
z 1 − z
k
n ∈ N
; |z| 1 .
Generating functions involving the Stirling numbers 207
2.6. Generalized Sylvester Polynomials
For the generalized Sylvester polynomials ϕ
n
x ; c defined by [14], p. 450, Problem
20 iv ϕ
n
x ; c : =
cx
n
n
2
F −n, x;
; − 1
cx = −1
n
L
−x−n n
cx in terms of the classical Laguerre polynomials [15], Chapter 5, it is known that [14], p.
450, Problem 20 v
∞
X
k=0
n + k k
ϕ
n+k
α ; x t
k
= 1 − t
−α−n
e
α x t
ϕ
n
α ; x 1 − t
n ∈ N
; |t| 1 , so that Theorem 1 immediately yields the generating function:
∞
X
k=0
k
n
ϕ
k
α ; x 1 + z
z 1 + z
k
= 1 + z
α
e
α x z
n
X
k=0
k S n, k ϕ
k
α ; x z
k
n ∈ N
; |z| 1 , which, for z 7−→ z 1 − z and x 7−→ x 1 − z, assumes the form:
∞
X
k=0
k
n
ϕ
k
α ; x z
k
= 1 − z
−α
e
α x z
·
n
X
k=0
k S n, k ϕ
k
α ; x 1 − z
z 1 − z
k
n ∈ N
; |z| 1 .
2.7. Bessel Polynomials
The Bessel polynomials y
n
x , α, β are defined by [14], p. 75, Equation 1.9 1 y
n
x , α, β : =
n
X
k=0
n k
α + n + k − 2 k
k x
β
k
=
2
F −n, α + n − 1;
; − x
β =
− x
β
n
n L
1−α−2n n
β x
208 Shy-Der Lin - Shih-Tong Tu - H. M. Srivastava
and satisfy the generating-function relationship [14], p. 419, Equation 8.4 8:
∞
X
k=0
y
n+k
x , α − k, β
t
k
k 18
= 1 −
x t β
1−α−n
e
t
y
n
x 1 −
x t β
−1
, α, β n
∈ N ; |t| |βx| .
On the other hand, for the simple Bessel polynomials y
n
x defined by y
n
x : = y
n
x , 2, 2 , it is known that [14], p. 419, Equation 8.4 10
∞
X
k=0
y
n+k
x t
k
k = 1 − 2xt
−
1 2
n+1
19 · exp
x
−1
h 1 −
√ 1 − 2xt
i y
n
x √
1 − 2xt n ∈ N
; |t| 1
2 |x|
−1
. Thus, in view of the obviously independent results 18 and 19, Theorem 3 yields the
following presumably new generating functions for the Bessel polynomials:
∞
X
k=0
k
n
k y
k
x 1 +
x z β
−1
, α − k, β
z
k
= 1 +
x z β
α −1
e
z n
X
k=0
S n, k y
k
x , α − k, β z
k
n ∈ N
; |z| |βx| , which, for
x 7−→ x 1 −
x z β
−1
, assumes the form:
∞
X
k=0
k
n
k y
k
x , α − k, β z
k
= 1 −
x z β
1−α
e
z
·
n
X
k=0
S n, k y
k
x 1 −
x z β
−1
, α − k, β
z
k
n ∈ N
; |z| |βx| ;
Generating functions involving the Stirling numbers 209
∞
X
k=0
k
n
k y
k
x √
1 + 2x z z
√ 1 + 2x z
k
= √
1 + 2x z exp −x
−1
h 1 −
√ 1 + 2x z
i ·
n
X
k=0
S n, k y
k
x z
k
n ∈ N ; |z|
1 2
|x|
−1
.
2.8. Generalized Heat Polynomials
