The concept of exponentially convex functions was introduced by S. N. Bernstein and D. V. Widder independently. This concept is extended in this paper for the matrix-valued functions and sequences; moreover we deal with the Hankelian totally positive (non-negative, matrix-valued) functions and sequences. The paper consists of six parts. In the first one we introduce the usual concepts and notations. The subject of the second part is the positive definite (semidefinite) hypermatrices of grade \(p\). We prove among others an extension of the well-known Lagrange transformation. The third part contains an appropriate generalization of the Landsberg theorem, which has an importance in the theory of total positivity. Using the results of the third part we give a procedure to construct totally positive matrices, and hypermatrices of grade \(p\), respectively, in the fourth part. After these preparatory parts we deal with matrix-valued exponentially convex functions, with matrix-valued Hankelian totally positive functions, and with matrix-valued absolutely monotone functions in the fifth part. At the same place we give characterizations for these matrix-valued functions, and also some relations among them. The last part contains a generalization of the Hamburger moment problem for matrix-valued sequences, as well as a new proof of the full Stieltjes moment problem. The results of the paper can be applied in several fields of probability theory, for example in the theory of birth and death processes. We return to these questions in another paper.