New sums mixing harmonic numbers and central binomial coefficients

Technical Summary: New Sums Mixing Harmonic Numbers and Central Binomial Coefficients

Problem Statement
The paper addresses the evaluation of combinatorial sums involving reciprocals of central binomial coefficients, specifically those mixed with harmonic numbers ($H_n$), odd harmonic numbers ($O_n$), and generalized harmonic numbers of order 2 ($H_n^{(2)}$). While identities for sums of the form $\sum \frac{2^{2k}}{\binom{2k}{k}}$ and their generalizations are well-documented (e.g., by Rockett, Comtet, and Gould), the authors focus on deriving closed-form expressions and recursive solutions for sums where these coefficients are multiplied by powers of the index ($k^d$) and harmonic terms. The work extends previous findings by Bataille and Frontczak regarding generalized binomial sums to include harmonic components.

Methodology
The primary methodology employed is the derivative operator approach applied to known combinatorial identities. The authors utilize the relationship between harmonic numbers and the digamma function, $\psi(z)$, where $H_z = \psi(z+1) + \gamma$. By differentiating existing identities with respect to a parameter $m$ (where the binomial coefficient is generalized to $\binom{m+k}{k}$), they generate terms involving $H_{k+m}$.

Key technical steps include:

1. Differentiation of Generalized Identities: Starting from the identity established in [3, Theorem 4.2]:
   $$\sum_{k=0}^n \frac{2^{2k}}{\binom{2k}{k}} \binom{m+k}{k} = \frac{1}{2m+3} \left( (m+n+1)\binom{m+n}{n}\binom{2n}{n} 2^{2n+1} + 1 \right)$$
   The authors differentiate this with respect to $m$, utilizing the property $\frac{d}{dm}\binom{m+k}{k} = \binom{m+k}{k}(H_{k+m} - H_m)$.
2. Summation by Parts (Abel's Transformation): For the power sums $U_d(n)$ and $V_d(n)$, the authors employ Abel's transformation to derive recursive relations. They define auxiliary sequences $u_k = \frac{2^{2k}}{\binom{2k}{k}}$ and $a_k = (2k-1)(k-1)^d$ to manipulate the sums into recursive forms involving lower-order sums.
3. Algebraic Simplification: The resulting expressions are simplified by isolating rational and irrational parts (specifically separating terms involving $\ln 2$ when dealing with half-integer arguments) and utilizing known relations between $H_n$, $O_n$, and $H_{2n}$.

Key Contributions and Results

1. Generalized Harmonic Sums (Theorem 2.1):
   The paper establishes a general identity for sums of the form $\sum_{k=0}^n \frac{2^{2k}}{\binom{2k}{k}} \binom{m+k}{k} H_{k+m}$. This result is valid for complex $m$ (excluding non-positive integers and $-3/2$).

   • Corollary 2.2 & 2.4: By setting specific values for $m$, the authors derive new closed-form expressions for:
     $$\sum_{k=0}^n \frac{2^{2k}}{\binom{2k}{k}} H_k \quad \text{and} \quad \sum_{k=0}^n \frac{2^{2k}}{\binom{2k}{k}} H_{2k}$$
     These are presented as potentially new results, distinct from previously known identities for odd harmonic numbers.

2. Recursive Solutions for Power Sums (Theorems 3.1 & 3.3):
   The authors define two classes of sums:

   • $U_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} k^d$
   • $V_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} k^d H_k$
     They derive recursive formulas that express $U_d(n)$ and $V_d(n)$ in terms of lower-order sums ($U_{d-j}$ and $V_{d-j}$) and boundary terms. Explicit closed forms are provided for $d=1, 2, 3$.
   • Observation: The paper notes that $V_d(n)$ can be expressed in a specific polynomial form involving $H_n$, a constant term, and a normalizing factor $N(d) = \prod_{j=0}^{d+1} (2j+1)$.

3. Second Family of Harmonic Sums (Theorem 4.1):
   A second family of identities is derived involving the term $\frac{2^{-2k}}{2k-1}$. Theorem 4.1 provides a general formula for sums involving $\binom{m+k+1}{k} H_{m+k+1}$. Corollaries provide specific evaluations for cases where $m=0$ and $m=1$, relating these sums to $H_{n+1}$ and $H_{n+2}$.

4. Sums Involving Squared Harmonic Numbers (Theorem 5.1):
   The paper investigates sums involving $H_k^2 - H_k^{(2)}$ (where $H_k^{(2)} = \sum_{j=1}^k j^{-2}$). By differentiating the identity in Theorem 2.1 twice (or utilizing the derivative of the digamma function), they derive a closed form for:
   $$\sum_{k=0}^n \frac{2^{2k}}{\binom{2k}{k}} \binom{m+k}{k} (H_{k+m}^2 - H_{k+m}^{(2)})$$

   • Corollary 5.2: Specific results for $\sum \frac{2^{2k}}{\binom{2k}{k}} H_k^2$ and $\sum \frac{2^{2k}}{\binom{2k}{k}} H_k^{(2)}$ are provided.
   • Limitation: The authors explicitly state that a closed-form evaluation for the residual sum $\sum_{j=1}^n \frac{2^{2j}}{j^2 \binom{2j}{j}}$ remains elusive and was not found in existing literature.

Significance and Scope
The paper positions itself as an addendum to previous work by the authors [3], extending the scope of known identities for central binomial coefficients to include harmonic numbers and their powers. The significance lies in:

• Providing new closed-form identities for sums mixing central binomial coefficients with $H_k$ and $H_{2k}$.
• Establishing systematic recursive methods to compute sums of the form $\sum \frac{2^{2k}}{\binom{2k}{k}} k^d H_k$ for any positive integer $d$.
• Highlighting the utility of the derivative operator method in generating new combinatorial identities from existing ones.

The authors maintain a modest tone, noting that while the derived polynomials $P_d(n)$ and $Q_d(n)$ in the recursive solutions are interesting, their deeper structural properties warrant further study. They also suggest future research directions involving generating functions with variable $x$ (analogous to work by Mansour and Batir/Sofo) to better understand the polynomials associated with these sums. The paper does not propose experimental applications but focuses strictly on the theoretical derivation and evaluation of these combinatorial sums.