On identities involving generalized harmonic , hyperharmonic and special numbers with Riordan arrays


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                        <jats:tex-math>\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)} ,</jats:tex-math>
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                  </jats:disp-formula>and for <jats:italic>n</jats:italic> > <jats:italic>r</jats:italic> ≥ 0,
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                        <jats:tex-math>\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)} ,</jats:tex-math>
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               <jats:p>where Bernoulli numbers <jats:italic>Bn</jats:italic> and Stirling numbers of the first kind <jats:italic>s</jats:italic> (<jats:italic>n</jats:italic>, <jats:italic>r</jats:italic>).</jats:p>


Introduction
The harmonic numbers Hn = n k= k frequently arise in combinatorial problems and in various branches of number theory. These numbers have been generalized by several authors. One of them is the generalized harmonic numbers Hn(α) [5] de ned by, for every ordered pair (α, n) ∈ R + × N, In [9], the generalized hyperharmonic numbers of order r, H r n (α) are de ned by In [3], the generalized harmonic numbers of rank r, H (n, r, α) are de ned as for n ≥ and r ≥ , H (n, r, α) = ≤n +n +...+nr≤n n n ...nr α n +n +...+nr or equivalently, as The generating function of the generalized harmonic numbers of rank r, H (n, r, α) is given by For α = , H (n, r, ) = H (n, r) were introduced in their works [6,10]. x n stands for the falling factorial de ned by x n = x (x − ) . . . (x − n + ) for n ≥ and x = .
It is well known that Stirling numbers play an important role in combinatorial analysis. The Stirling numbers of the rst kind s(n, k) and of the second kind S (n, k) are given by respectively. The signed Stirling numbers of the rst kind s(n, k) are de ned such that the number of permutations of n elements which contain exactly k permutation cycles is the nonnegative number This means that s(n, k) = for k > n and s(n, n) = and the generating function of these numbers [1,7,15] is given by The numbers associated with s(n, k) are given as follows: For n < k, where σn (x) is Stirling polynomial [7]. The generating function of these numbers is Recently, by using the concepts of Riordan arrays, there are some identities related to special numbers and binomial coe cients [1,13,14]. Let f (x) be a formal power series in the determinate. f (x) has the form Riordan array is an in nite, lower triangular array R = (g (x) , f (x)) = r n,k n,k≥ de ned by a pair of generating functions g(x) and f (x) such that where x n denotes the coe cient of x n in f (x) . If g ( ) ≠ , f ( ) ≠ , Riordan array is called proper, otherwise it is called improper. The set of proper Riordan arrays is denoted by R and the set of improper Riordan arrays is denoted by R . It is known [11] that R, * forms a group under matrix multiplication * with the identity I = ( , ) : Basically, the concept of a Riordan array is used in a constructive way to nd the generating function of many combinatorial sums. The summation property for a Riordan array R = (g (x) , f (x)) = r n,k n,k≥ is where h (x) = ∞ n= hn x n . Riordan arrays have special structure. If R = (g (x) , f (x)) = r n,k n,k≥ is a proper Riordan array, then every element r n+ ,k+ of R can be expressed as a linear combination of the elements in preceding row starting from the preceding column, and every element in column can be expressed as a linear combination of all elements of the preceding row [8]:  cients a , a , a , ... and z , z , z , ... are called by the A−sequence and Z−sequence of Riordan array, respectively. If A (x) and Z (x) are the generating functions of corresponding sequences, then it can be proven that f (x) and g (x) are solutions of the functional equations, respectively: .
The relations can be inverted to formulas for the A−sequence and Z−sequence, respectively. From special cases, it is seen that the Riordan array method is powerful for dealing with harmonic numbers identities.
In [13], the author obtained some identities involving binomial coe cients and harmonic numbers by Riordan arrays. For example, for n ≥ , In [14], the author established several general summation formulas, from which series of harmonic number identities are obtained. For example, In [2], the authors established some new identities involving Stirling numbers of both kinds. These identities were obtained via Riordan arrays with nonzero real number x. Some well-known identities were obtained as special cases of the new identities for nonzero real number

Some identities with Riordan arrays
In this section, we establish some identities related to generalized harmonic numbers of rank r, H (n, r, α) , generalized hyperharmonic numbers H r n (α) and Stirling numbers of both kinds via Riordan arrays. Firstly, by (1.3), we can consider Thus, to nd the following identities in Theorem 1 and Theorem 2, we will apply the summation property (1.10) to the Riordan array.
and from (1.1), equals Thus, the proof of the rst equality is complete. With the help of (1.4), the other equality is easily given. x ∈ R. Thus, to nd the following identities in Theorem 3, we will apply the summation property (1.10) to the Riordan array. From (1.2), we have r n,k = H k n−k (α) and R = − ln − x α , −x ∈ R . Hence, the following theorem is clearly given. Thus, from aboving, the proof is complete.