Addition Formulas for Jacobi, Gegenbauer, Laguerre, and Hyperbolic Bessel Functions of the Second Kind

We present a set of addition formulas for the Jacobi, Laguerre, Gegenbauer, and hyperbolic Bessel functions of the second kind, $Q_\nu ^{(\alpha ,\beta )} $, $N_\nu ^\alpha $, $D_\nu ^\alpha $ and $K_\nu $. These addition formulas are analogues of Koornwinder’s addition formulas for $P_n^{(\alpha ,\beta )} $ and $L_n^\alpha $, and of Gegenbauer’s addition formulas for $C_n^\alpha $ and $J_n $. The addition formulas are derived from a set of product formulas for the functions of the second kind derived previously by the author, and, conversely, can be integrated to give the product formulas.