The aim in this research note is to provide an extension of Saalschütz's summation theorem for the series r+3Fr+2(1) when r pairs of numeratorial and denominatorial parameters differ by positive integers. The result is obtained by exploiting a generalization of an Eulertype transformation recently derived by Miller and Paris [9].

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For time-to-event data, the power of the two sample logrank test for the comparison of two treatment groups can be greatly influenced by the ratio of the number of patients in each of the treatment groups. Despite the possible loss of power, unequal allocations may be of interest due to a need to collect more data on one of the groups or to considerations related to the acceptability of the treatments to patients. Investigators pursuing such designs may be interested in the cost of the unbalanced design relative to a balanced design with respect to the total number of patients required for the study. We present graphical displays to illustrate the sample size adjustment factor, or ratio of the sample size required by an unequal allocation compared to the sample size required by a balanced allocation, for various survival rates, treatment hazards ratios, and sample size allocation ratios. These graphical displays conveniently summarize information in the literature and provide a useful tool for planning sample sizes for the two sample logrank test. SUMMARYFor time-to-event data, the power of the two sample logrank test for the comparison of two treatment groups can be greatly influenced by the ratio of the number of patients in each of the treatment groups.Despite the possible loss of power, unequal allocations may be of interest due to a need to collect more data on one of the groups or to considerations related to the acceptability of the treatments to patients.Investigators pursuing such designs may be interested in the cost of the unbalanced design relative to a balanced design with respect to the total number of patients required for the study. We present graphical displays to illustrate the sample size adjustment factor, or ratio of the sample size required by an unequal allocation compared to the sample size required by a balanced allocation, for various survival rates, treatment hazards ratios, and sample size allocation ratios. These graphical displays conveniently summarize information in the literature and provide a useful tool for planning sample sizes for the two sample logrank test.

By elementary manipulation of series, a general transformation involving the generalized hypergeometric function is established. Kummer's first theorem, the classical Gauss summation theorem and the generalized Kummer summation theorem due to Lavoie, Grondin and Rathie [J. Comput. Appl. Math. 72 (1996) 293-300] are then applied to obtain a new class of summation formulae involving the Laguerre polynomial, which have not previously appreared in the literature. Several related results due to Exton have also been given in corrected form.

A reactive hot melt adhesive (RHMA) consisting of thermoplastic polyurethane (TPU) was modified with sodium montmorillonite (Na-MMT) intercalated with poly(ethylene glycol) (PEG), and their effects on the adhesion, rheological, and mechanical properties of the RHMA were examined. The Na-MMT intercalated with PEG (Na-MMT/PEG) effectively enhanced the initial bond strength development of the RHMA, although the amounts of Na-MMT/PEG in the RHMA were less than 0.2%. The increase of the complex viscosity and pseudo-solid like behavior observed at low shear rate indicates that there are intimate interactions between the RHMA molecules and Na-MMT/PEG. The improved modulus and tensile strength of the cured RHMA film in the presence of Na-MMT/PEG demonstrates that Na-MMT/PEG effectively reinforced the RHMA.