Harmonic Series with Multinomial Coefficient <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mrow><mn>4</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>n</mi></mrow></mfrac></mfenced></semantics></math></inline-formula> and Central Binomial Coefficient <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mrow><mn>2</mn><mi>n</mi></mrow><mi>n</mi></mfrac></mfenced></semantics></math></inline-formula>

Abstract

Classical hypergeometric series are reformulated as analytic functions of their parameters (in both the numerator and the denominator). Then, the coefficient extraction method is applied to examine hypergeometric series transformations. Several new closed form evaluations are established for harmonic series containing multinomial coefficient 4nn,n,n,n and central binomial coefficient 2nn. These results exclusively concern the alternating series of convergence rate “−1/4”.

Keywords

• hypergeometric series
• harmonic number
• Riemann zeta function
• multinomial coefficient