XVII.—Theorems relating to a Generalization of Bessel's Function

Extract

The theory of the functions commonly known as q functions might perhaps be greatly developed, if investigators were to work on lines suggested by the functional notation of well-known analytie functions. For instance, the analysis connected with the circular functions sin x, cos x, …, might be regarded as the theory of certain infinite products without using any special functional notation. It need not be explained however, how great was the gain to elementary algebra by the introduction of the exponential function (regarded as the limit of a certain infinite product, or as the limit of a certain infinite series) denoted e^x, with certain characteristic properties, enabling the worker to make transformations easily and quickly. Of course, the vast store of interesting and in many cases useful results connected with the elementary functions of analysis might have been obtained without the introduction of any notation capable of rapid and easy transformations, but I think it unlikely that they would have been obtained.