Abstract
In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0,
∑
k
=
0
n
B
k
k
!
H
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n
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k
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α
)
=
α
H
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n
+
1
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1
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α
)
-
H
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n
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1
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,
\sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} ,
and for n > r ≥ 0,
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r
n
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1
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-
1
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k
s
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k
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r
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r
!
α
k
k
!
H
n
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k
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=
(
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1
)
r
H
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n
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r
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,
\sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} ,
where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).
$q$-Riordan array for $q$-Pascal matrix and its inverse matrix