Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant †
Abstract
:1. Preliminaries and Motivations
- Derive closed-form expressions for partial Bell polynomials at specific arguments, where these closed-form formulas were asked for in [25];
- Establish closed-form formulas for partial Bell polynomials at specific arguments of the sequences
2. Closed-Form Formulas for Partial Bell Polynomials
- If , we acquire a combinatorial identity:
- If for , we have
- If for , we have
3. Power Series Expansions for and
4. Power Series Expansions for and
5. Infinite Series Representations for and Its Powers
6. Several Combinatorial Identities
7. Useful Remarks
- https://math.stackexchange.com/a/4157803 (accessed on 12 April 2024);
- https://math.stackexchange.com/a/4379999 (accessed on 13 March 2023);
- https://math.stackexchange.com/a/4380027 (accessed on 13 March 2023);
- https://math.stackexchange.com/a/4657809 (accessed on 12 March 2023).
8. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Qi, F. Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant. Symmetry 2024, 16, 1145. https://doi.org/10.3390/sym16091145
Qi F. Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant. Symmetry. 2024; 16(9):1145. https://doi.org/10.3390/sym16091145
Chicago/Turabian StyleQi, Feng. 2024. "Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant" Symmetry 16, no. 9: 1145. https://doi.org/10.3390/sym16091145