Gauss’ Second Theorem for <inline-formula><math display="inline"><semantics><mrow><mmultiscripts><mi>F</mi><mn>1</mn><none/><mprescripts/><mn>2</mn><none/></mmultiscripts><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-Series and Novel Harmonic Series Identities

Abstract

Two summation theorems concerning the F12(1/2)-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate “1/2” may serve as a reference source for readers.

Keywords

• harmonic number
• gamma function
• binomial/multinomial coefficient