The authors develop interesting algorithms for such type of identities that involve multiple sums and integrals of products of special functions of hypergeometric type. The ideas of rational function certification, WZ- pair, and companion identity introduced by the authors in their earlier works [J. Am. Math. Soc. 3, No. 1, 147-158 (1990; Zbl 0695.05004
); Bull. Am. Math. Soc., New Ser. 23, No. 1, 77-83 (1990; Zbl 0718.05010
)] are extended to multiple sums and integrals. The algorithm provability is also stated for the $q$-hypergeometric identities. An explicit example of the algorithm depicts the computer proof of the well-known Hille-Hardy bilinear formula for Laguerre polynomials.