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Example 5. Here are some examples for the above case

H3(1r(2);1r(2);2)=ζ(2)ζ(3)=π26ζ(3) \begin{aligned} \mathcal{H}^3 (1_{r(2)}; 1_{r(2)}; 2) &= \zeta(2) - \zeta(3) \\ &= \frac{\pi^2}{6} - \zeta(3) \end{aligned}

H4(1r(2);1r(2);3)=ζ(3)ζ(4)=ζ(3)π490 \begin{aligned} \mathcal{H}^4 (1_{r(2)}; 1_{r(2)}; 3) &= \zeta(3) - \zeta(4) \\ &= \zeta(3) - \frac{\pi^4}{90} \end{aligned}

2 Conclusion

This paper was just an introduction to Hyder Series. We came to see some important results and some special cases of Hyder series of higher order. This series is named in honour of my late Grandfather Syed Hyder, who was a chief executive engineer in the water department of state Tamil Nadu. He enjoyed solving mathematical problems during his leisure times and was a man of wit and humour. I hope this Series could have many more other interesting result to be discovered and could also have unique relation to some special functions.

References

[1] Cornel Ioan Vălean. (Almost) impossible integrals, sums, and series. Springer, 2019.

[2] Eric W Weisstein. Euler-mascheroni constant. https://mathworld.wolfram.com/COMS.html, 2002.

[3] Eric W Weisstein. Hypergeometric function. https://mathworld.wolfram.com/COMS.html, 2002.

[4] Eric W Weisstein. Lerch transcendent. https://mathworld.wolfram.com/COMS.html, 2002.

[5] Eric W Weisstein. Polygamma function. https://mathworld.wolfram.com/COMS.html, 2002.