From 1942 until 1946 Luke served in the U.S. navy, being stationed in Hawaii. After his war service ended in 1946, he returned to Kansas City, Missouri with his wife LaVerne Podoll, who was from Chicago, and the two children which they had at that time. Yudell and Laverne Luke had two more children making a total of four girls.
Luke was appointed to the Midwest Research Institute soon after he returned to Kansas City in 1946. His first appointment was as Head of the Mathematical Analysis Section, a position he held until he was made Senior Advisor for Mathematics in 1961. Promotion to Senior Advisor in Mathematics in 1967 was only to last until 1971 for at that time the mathematics group at the Midwest Research Institute was disbanded. At the Institute :-
... in addition to his own research activities and the supervision of the research members of the Applied Mathematics Group, Professor Luke had a variety of responsibilities including that of procuring research projects for the Institute from government and industrial organisations. Several of his research students from this period matured into well-known research mathematicians who often collaborated with him. Professors J Wimp, W Fair and J L Fields are three of his best known former research students.However, his posts at the Midwest Research Institute were not the only ones he held. In 1955 Luke had been appointed a lecturer at the University of Missouri in Kansas City. He also taught at the University of Kansas and, after the mathematics group at the Midwest Research Institute was disbanded in 1971, Luke was appointed as professor at the University of Missouri in Kansas City. In 1975 he was honoured by the University of Missouri with the award of the N T Veatch award for Distinguished Research and Creative Activity. Then, in 1978, he was honoured with the appointment as Curator's Professor at the University of Missouri, a post he held until his death.
Luke published nearly 100 papers and eight books during his highly distinguished career. This work falls into a number of different areas but it began with applied mathematics and research into aeronautics. In this area he published on the forces on aircraft wings, in particular studying stress and sonic flutter. His work on these topics led him to require much information on special functions and he was led to develop tables of special functions and to use numerical techniques to solve equations. His early work on Bessel functions and hypergeometric functions appeared in his first major text Integrals of Bessel functions which was published in 1962.
In order to compute tables of special functions, Luke needed to acquire expertise in approximation theory and in this way he was led to the main area of research on which he was to become a leading world expert. This is explained in  as follows:-
He was one of the first mathematicians to realise the potential of the Tau Method for the analysis and praxis of numerical approximation problems. This approach, due to Cornelius Lanczos, is based on the ideas of best uniform approximation by polynomials and rational functions. Luke used this method at a time when most of the interest in numerical analysis was still centred around finite difference techniques.Not only did he use rational approximation, but Luke also developed series expansions as an approximation method. For example he expanded hypergeometric functions in series of Laguerre and Hermite polynomials. Many of these methods involved great computational problems and Luke was led to another important area of his research, namely the design of algorithms to implement his numerical approximations. Some of his books record his great research achievements. For example The special functions and their approximations (1969) and two further volumes Mathematical functions and their approximation (1975) and Algorithms for the computation of mathematical functions (1977) contain a beautiful survey of the areas on which he worked. These texts are described in  as follows:-
These works contain an amazing wealth of information, theoretical as well as practical, pertaining to special functions, summarising and systemising to a large extent Yudell's own research and that of his collaborators, without neglecting, however, relevant work of others.It was not only through his research, however, that Luke contributed to mathematics. He was an industrious reviewer, reviewing by his own estimation over 1800 papers and books throughout his career. He received awards from Applied Mechanics Reviews in both 1972 and 1981 for his outstanding service. Another of his interests was in classifying information and he made substantial contributions to this in his work in preparing a cumulative index for the first 23 volumes of the Mathematics of Computation.
A keen supporter of various mathematical societies, we should mention in particular his efforts in setting up the Visiting Lecturer Program for the Society of Industrial and Applied Mathematics. He, himself, was a visiting lecturer for the Society in 1960-61, 1964-65 and 1975-76.
In 1982 an exchange programme between the University of Missouri in Kansas City and the University of Moscow was set up. In 1983 Luke travelled to Moscow to lecture there as part of this exchange programme. He gave a wonderful series of lectures on special functions, asymptotic analysis, and approximation theory. Tragically, however, he died while still in Moscow.
His interests outside mathematics are described in :-
Playing bridge and cribbage and participating in baseball and basketball were his favorites.In fact Luke wrote two books on the probabilities of winning at the card game of cribbage.
His interests are also described in :-
He loved opera, philosophy, baseball, among other things. While at MRI he gave an extensive series of lectures on the history of philosophy, focusing especially on Spinoza, whose work he believed, contains the most meaningful elements of those ethical and intellectual ideals which alone can provide a personal bedrock in an uncertain, frenetically changing world.One of his four daughters wrote :-
He was a very generous person and his religion meant a great deal to him. ... He was very special to me and I would like for everybody to know what a wonderful man he was. His wife, LaVerne, is still living in Kansas City. He has two daughters in Kansas City, one daughter in California, and one daughter in Florida. There are eight grandchildren all over the country who were very close to him.
Article by: J J O'Connor and E F Robertson