TOTAL VIEWS: 147

Published: September 30,2024

In this note, we employ two identities from symbolic calculations and combine several known series expansions for specific special functions. By utilizing these tools, we can generate numerous Euler-like sums that connect harmonic numbers, pi and zeta values etc. These results are aesthetically pleasing and widely used in Euler-like sums fields. The main step is to calculate some new Euler-like sums of the harmonic number by integration transformations. The definite integrals of some elementary functions are first calculated by symbol software Mathematica. On the other hand, we can write these elementary functions in the form of Maclaurin series and then integrate them term by term and finally get the series containing harmonic numbers. In the process, we need to deal with the series, integrals, differentiating and other operations skillfully. We can continue to use the integral transform in this paper to expand, or we can find another new integral transform to get more valuable results by using similar methods.

[1] Campbell, John M., and Anthony Sofo. An integral transform related to series involving alternating harmonic numbers. Integral Transforms and Special Functions, 28.7 (2017): 547-559.

[2] Campbell JM. Ramanujan-like series for π 1 involving harmonic numbers, and related integration results. HAL: 01364815v1; 2016.

[3] Nimbran, Amrik Singh, Paul Levrie, and Anthony Sofo. Harmonic-binomial Euler-like sums via expansions of (\arcsin x)^ p (arcsin x) p. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 116 (2022): 1-23.

[4] Chen H. Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers. J Integer Seq., 2016; 19: Article 16.1.5.

[5] Sofo A. Derivatives of Catalan related sums. J Inequal Pure Appl Math., 2009; 10(3): Article 69.

[6] Janous W. Around Apéry’s constant. J Inequal Pure Appl Math., 2006; 7(1): Article 35.

[7] Sofo A. New families of alternating harmonic number sums. Tbilisi Math J., 2015; 8(2):195-209.

[8] Flajolet, Philippe, and Bruno Salvy. Euler sums and contour integral representations. Experimental Mathematics, 7.1 (1998): 15-35.

[9] Sofo, Anthony. Evaluating log-tangent integrals via Euler sums. Mathematical Modelling and Analysis, 27.1 (2022): 1-18.

[10] Adamchik, Victor. On Stirling numbers and Euler sums. Journal of Computational and Applied Mathematics, 79.1 (1997): 119-130.

[11] Guo, Dongwei. Some combinatorial identities concerning harmonic numbers and binomial coefficients. Discret. Math. Lett 8 (2022): 41-48.

[12] Sofo, Anthony, and Amrik Singh Nimbran. Euler sums and integral connections. Mathematics, 7.9 (2019): 833.

[13] Sebah, Pascal, and Xavier Gourdon. Introduction to the gamma function. American Journal of Scientific Research, (2002): 2-18.

**The Euler-like Sums Involving Harmonic Numbers Obtained from Integral Transformations**

**How to cite this paper:** Xuefeng Zhou. (2024) The Euler-like Sums Involving Harmonic Numbers Obtained from Integral Transformations. *Journal of Applied Mathematics and Computation*, **8**(**3**), 218-226.

DOI: https://dx.doi.org/10.26855/jamc.2024.09.004

- The Euler-like Sums Involving Harmonic Numbers Obtained from Integral Transformations
- Optimization Design and Experimental Research of Composite Material Bases for Marine Applications
- Discontinuous Galerkin Method of the Second Order Parabolic Differential Equation
- Effect of Physical Parameters and Tilt Angle on Nusselt Number
- Reachable Set Estimation for Neutral Markovian Jump Systems with Time-varying Delay and Distributed Delay
- On the Geometric Boundary of Some Convex Sets