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Abstract: In this paper we prove the transcendence of the solutions of the q-Painlevé equation of type $${A_6^{(1)}}$$. The q-Painlevé equation of type $${A_6^{(1)}}$$ is also called d-P II or q-P II and has a continuous limit to the Painlevé equation of type II. Its symmetry is (A 1 + A 1)(1).

Abstract: In this paper we prove that solutions of q-P(A 7) are all transcendental over C(t). We also investigate transcendence of solutions of q-P(A 6) and prove transcendence of hypergeometric solutions of q-P(A 6).

Pervaporation and sorption behavior of water-ethanol mixtures was investigated on block copolymers of polydimethylsiloxane (PDMS)-polysulfone (PS) and of PDMS-PS-poly(4-hydroxystyrene) (PHS) as well as on PDMS. The preferential permeation of ethanol through PDMS membranes was attributed to both the preferential sorption of ethanol and rather small differences in the diffusivities between ethanol and water components (6.4×10-6and 14×10-6cm2s-1, respectively, at 25°C and an ethanol content in the feed (xE) of 10 wt%). The block copolymers were considered to have microphase-separated structures, of which the continuous PDMS phase practically contributed to the permeation. The high sorption in the block copolymers was mainly attributed to the disperse phases and contributed little to the permeation. PDMS-PS-PHS block copolymers having PDMS weight fraction (wPDMS) of 0.6–0.7 showed permeabilities as high as commercial silicone rubber (wPDMS=0.7) and a little lower selectivity of pervaporation (ap=7.7 at xE=10 wt%) as compared with that of PDMS (ap=8.8–9.4). They were superior to PDMS-PS block copolymers (wPDMS=0.5–0.6) as for both the selectivity and the permeability. Composite membranes comprising a uniform ultrathin layer of a PDMS-SP-PHS block copolymer on a microporous support had fairly high performance (the permeation rate of 4.5 kg m-2h-1and ap=6.8) and durability in the pervaporation of a dilute ethanol solution.