Letûðt; xÞ ¼ P 1 n¼1 u n ðxÞt n be a formal power series in one variable t satisfying formally Pðt; x; @ t ; @ x Þuðt; xÞ ¼ f ðt; xÞ; ðt; xÞ 2 C Â C d : The purpose of the present paper is to introduce a class of linear partial differential operators and show that uðt; xÞ is multisummable provided Pðt; x; @ t ; @ x Þ belongs to this class.

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A functional equation P m i¼1 a i uðj i ðzÞÞ ¼ f ðzÞ is considered, where fj i ðzÞg m i¼1 are holomorphic functions in a neighborhood of z ¼ 0 with j i ð0Þ ¼ 0 and f ðzÞ is holomorphic in a sector with vertex z ¼ 0. It is shown under some conditions of fj i ðzÞg m i¼1 , f ðzÞ and fa i g m i¼1 that the equation has a formal power series solution that is Borel summable.