Certain Applications of Generalized Kummer’s Summation Formulas for 2F1

We present generalizations of three classical summation formulas 2F1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number of interesting formulas for double-finite series involving quotients of Gamma functions. We also consider several other applications of these formulas. Certain symmetries occur often in mathematical formulae and identities, both explicitly and implicitly. As an example, as mentioned in Remark 1, evident symmetries are naturally implicated in the treatment of generalized hypergeometric series.

Both Classical and Quantum Information: Both Bit and Qubit. Both Physical and Transcendental Time

Information can be considered as the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information, particularly by their units, bit and qubit, correspond and unify the finite and infinite. As classical information is relevant to finite series and sets, as quantum information, to infinite ones. A fundamental joint relativity of the finite and infinite, of the external and internal is to be investigated. The corresponding invariance is able to define physical action and its quantity only on the basis of information and especially: on the relativity of classical and quantum information. The concept of transcendental time, an epoché in relation to the direction of time arrow can be defined. Its correlate is that information invariant to the finite and infinite, therefore unifying both classical and quantum information.

Both classical & quantum information: both bit & qubit. Both physical & transcendental time

Information can be considered as the most fundamental, philosophical,physical and mathematical concept originating from the totality by means of physicaland mathematical transcendentalism (the counterpart of philosophicaltranscendentalism). Classical and quantum information, particularly by their units, bitand qubit, correspond and unify the finite and infinite. As classical information isrelevant to finite series and sets, as quantum information, to infinite ones. Afundamental joint relativity of the finite and infinite, of the external and internal is tobe investigated. The corresponding invariance is able to define physical action and itsquantity only on the basis of information and especially: on the relativity of classicaland quantum information. The concept of transcendental time, an epoché in relation tothe direction of time arrow can be defined. Its correlate is that information invariant tothe finite and infinite, therefore unifying both classical and quantum information.

Finite Series of Distributional Solutions for Certain Linear Differential Equations

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations t2y″(t)+2ty′(t)−[t2+ν(ν+1)]y(t)=0 and the linear differential equations of the forms t2y″(t)+3ty′(t)−(t2+ν2−1)y(t)=0, where ν∈N∪{0} and t∈R. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of ν. The results of several examples are also presented.