There is not a specific version of lambda calculus known as a “hybrid multi-typed and polymorphic lambda calculus.” However, it is possible to imagine a version of lambda calculus that combines the features of both multi-typed and polymorphic lambda calculi.
In this hybrid version of lambda calculus, expressions could be assigned to multiple types, just as in polymorphic lambda calculus. However, the type of an expression could also change dynamically during the course of a computation, just as in multi-typed lambda calculus. This would allow for even greater flexibility and expressiveness than either polymorphic or multi-typed lambda calculus alone.
Such a hybrid version of lambda calculus would provide the benefits of both polymorphism and multi-typing, allowing for more flexible and expressive programs than a traditional typed lambda calculus. It would also allow for more complex and dynamic computations than either polymorphic or multi-typed lambda calculus alone.
However, it is important to note that there is no specific version of lambda calculus known as a “hybrid multi-typed and polymorphic lambda calculus.” This is simply a theoretical concept, and it may or may not have been explored in the research literature on lambda calculus.