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Some of our office employees will work remotely to secure business. Untill now we will remain fully operational.
This being said, we cannot ensure production and delivery in case the Local Authorities would announce nationwide quarantine, or the freight companies and carriers would be restricted.
Your browser does not support the video tag. CONTACT US. ABOUT MEIJER GROUP. Groot Verzet tegen Kanker verplaatst naar Again, the formula can be verified by expressing the two G-functions as sums of residues ; no cases of confluent poles permitted by the definition of the G-function need be excluded here.
As can be seen from the definition of the G-function , if equal parameters appear among the a p and b q determining the factors in the numerator and the denominator of the integrand, the fraction can be simplified, and the order of the function thereby be reduced.
Whether the order m or n will decrease depends of the particular position of the parameters in question. Concerning derivatives of the G-function, one finds these relationships:.
From these four, equivalent relations can be deduced by simply evaluating the derivative on the left-hand side and manipulating a bit.
One obtains for example:. From these relationships, corresponding properties of the Gauss hypergeometric function and of other special functions can be derived.
By equating different expressions for the first-order derivatives, one arrives at the following 3-term recurrence relations among contiguous G-functions:.
Similar relations for the diagonal parameter pairs a 1 , b q and b 1 , a p follow by suitable combination of the above. Again, corresponding properties of hypergeometric and other special functions can be derived from these recurrence relations.
The radii of convergence will be dependent on the value of z and on the G-function that is expanded. The expansions can be regarded as generalizations of similar theorems for Bessel , hypergeometric and confluent hypergeometric functions.
Among definite integrals involving an arbitrary G-function one has:. Note that the restrictions under which this integral exists have been omitted here.
It is, of course, no surprise that the Mellin transform of a G-function should lead back to the integrand appearing in the definition above.