scholarly journals The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point

1949 ◽  
Vol 67 (2) ◽  
pp. 461-461 ◽  
Author(s):  
Rudolph E. Langer
Keyword(s):  
Differential Equations ◽  
Special Reference ◽  
Turning Point ◽  
Second Order ◽  
Linear Differential Equations ◽  
Asymptotic Solutions ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

scholarly journals Composition sum identities related to the distribution of coordinate values in a discrete simplex

10.37236/1498 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
R. Milson
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Algebraic Methods ◽  
Algebraic Proof ◽  
Geometric Properties ◽  
Linear Differential ◽  
Exponential Formula ◽  
Ordinary Linear Differential Equations ◽  
General Method

Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first and second order, ordinary, linear differential equations. Regarding the first class, the corresponding identities amount to a proof of the exponential formula of labelled counting. The identities in the second class can be used to establish certain geometric properties of the simplex of bounded, ordered, integer tuples. We present three theorems that support the conclusion that the inner dimensions of such an order simplex are, in a certain sense, more ample than the outer dimensions. As well, we give an algebraic proof of a bijection between two families of subsets in the order simplex, and inquire as to the possibility of establishing this bijection by combinatorial, rather than by algebraic methods.

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