Generating Functions for Binomial Series Involving Harmonic-like Numbers
Abstract
:1. Introduction and Motivation
1.1. Harmonic Numbers
1.2. Coefficient Extraction
1.3. Three -Series
2. Series with the Central Binomial Coefficient in the Denominator
2.1. Series from
- Two numerical series fromTheir linear combination leads to the following identity conjectured by Sun [27] (Page 7):
- Two further numerical series from :These are refined identities of the following identity conjectured by Sun [27] (Page 7):
- Analogous evaluations detected by Sun [28] (Conjectures 10.62 and 10.63):
- Numerical series from :
- Numerical series from :
- Particularly for , we deduce two numerical series
2.2. Series from
2.3. Series from
3. Series with the Central Binomial Coefficient in the Numerator
3.1. Series from
3.2. Series from
3.3. Series from
4. Series Containing the Binomial Coefficient
4.1. Series from
4.2. Series from
4.3. Series from
5. Series Containing the Central Binomial Coefficient
5.1. Series from
5.2. Series from
5.3. Series from
6. Series Involving Trinomial/Binomial Quotient
6.1. Series from
6.2. Series from
6.3. Series from
7. Concluding Remarks
7.1. General Forms of Power Series Expansions
- Coefficient of in (general form of Section 2.1):
- Coefficient of in (general form of Section 3.2):
- Coefficient of in (general form of Section 4.3):
7.2. Linearly Combined Series
7.3. Further Contiguous Series
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Li, C.; Chu, W. Generating Functions for Binomial Series Involving Harmonic-like Numbers. Mathematics 2024, 12, 2685. https://doi.org/10.3390/math12172685
Li C, Chu W. Generating Functions for Binomial Series Involving Harmonic-like Numbers. Mathematics. 2024; 12(17):2685. https://doi.org/10.3390/math12172685
Chicago/Turabian StyleLi, Chunli, and Wenchang Chu. 2024. "Generating Functions for Binomial Series Involving Harmonic-like Numbers" Mathematics 12, no. 17: 2685. https://doi.org/10.3390/math12172685