On Expanding the Sine Function with Catalan Numbers: A Note on a Role for Hypergeometric Functions

The theory of hypergeometric functions is brought to bear on a problem—namely, that of obtaining a certain power series expansion involving the sine function that is inclusive of the Catalan sequence and which serves as a prelude to the calculation of other related series of similar type. A general formulation provides the particular result of interest as a special case, into which Catalan numbers are introduced as desired.