**Wolfgang Hahn**'s father was a secondary school teacher. Wolfgang was only a young boy during the years of World War I, 1914-1918, but he witnessed the major changes to German society in his school days following the end of the war. In his class were a number of Hohenzollern princes who, unlike Wolfgang, were not too bright. There is a nice story about one of these princes who, despite the best efforts of the mathematics teacher, could not understand a proof despite the teachers best efforts. Eventually the teacher said, "But Imperial Highness, I give you my word of honour that the evidence is true." The prince replied, "If only you had explained that in the first place."

In 1928 Hahn graduated from the Gymnasium at Potsdam and began his university studies of mathematics at the Friedrich-Wilhelm University in Berlin. There he was taught by some outstanding mathematicians including Erhard Schmidt, Issai Schur, Ludwig Bieberbach, and Robert Remak. Hahn attended Schur's seminar and Remak also attended. However, Hahn said that Remak always seemed to be asleep in the seminar but, despite this, he would suddenly appear to wake up and ask a highly relevant question or explain a difficult point with which others were struggling. Hahn also saw Remak going rowing on the river when he would shout to passers-by, "Mathematicians are all crazy, we are all a little crazy."

Like most German students at this time, Hahn did not spend all of his university career at one university. He spent two semesters at the University of Göttingen where he attended lectures by Richard Courant, Edmund Landau and Gustav Herglotz. While at Göttingen, he met Emmy Noether, Bartel van der Waerden, Emil Artin and Ernst Witt. Hahn said that he had attended a lecture by Carl Ludwig Siegel but had not understood it at all. He got to know Edmund Landau well at this time and was often invited to his home. Hahn said that Edmund Landau always talked to his dog in Hebrew. Landau often organised quiz evenings to which Hahn was invited.

After his year in Göttingen, Hahn returned to Berlin where he continued to undertake research advised by Schur. In January 1933 he passed the State examinations which entitled him to become a gymnasium teacher. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course removing those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Since he was not Jewish, Hahn was not directly affected by this law but, nevertheless, it did have consequences for him. He submitted his doctoral thesis *Die Nullstellen der Laguerreschen und Hermiteschen Polynome* Ⓣ and was examined by Schur and other examiners in July 1933. His doctoral studies had been made with advice from Schur and Remak. His friendship Edmund Landau at Göttingen also was a black mark against him in the eyes of those now in authority in Germany since all three of his influential teachers were Jews dismissed under the Civil Service Law.

Hahn spent the years from 1933 to 1940 as a school teacher. Most of these seven years were spent in Berlin but he did spend one year teaching in the town of Züllichau in Brandenburg. This was particularly significant since one of the pupils he taught during this year was Irmgard Pollack. They later married and had one son Gerold Hahn who went on the become a secondary school teacher of mathematics in Kassel. During these years as a school teacher Hahn published some research articles: *Bericht über die Nullstellen der Laguerreschen und Hermiteschen Polynome* Ⓣ (1935); *Über die Jacobischen Polynome und zwei verwandte Polynomklassen* Ⓣ (1935); *Über höhere Ableitungen von Orthogonalpolynomen* Ⓣ (1935); and *Über Orthogonalpolynome mit drei Parametern* Ⓣ (1939).

In 1940 Hahn was drafted into the Wehrmacht. He was sent with the invading German forces to Norway where 300,000 German troops remained stationed for the duration of the war. No wartime experience is pleasant and, particularly towards the end of the war when the Norwegian resistance became stronger, this was far from a holiday. Still it may have been one of the luckier postings for German troops. Perhaps it is also worth noting that the Allies were happy to see this large cohort of German troops kept away from the areas of active fighting. Hahn remained in Germany until the war ended in 1945 and, following the victory by the Allies, he was prevented from returning to Germany until 1946. Back in Berlin in 1946 he was not able to continuing in the teaching profession and for the following few years he had to do jobs involving hard physical labour in industrial plants. Although in many ways this was hard, yet it did allow him to continue his mathematical research working towards habilitation.

Hahn published three papers in 1949, namely: *Über Orthogonalpolygnome, die q-Differenzengleichungen genügen* Ⓣ; *Über Orthogonalpolynome, die gleichzeitig zwei verschiedenen Orthogonalsystemen angehören* Ⓣ; and *Beiträge zur Theorie der Heineschen Reihen* Ⓣ. Then, in 1950, he submitted his habilitation thesis *Über höheren Heineschen Reihen und eine einheitliche* *Theorie der sogenannten speziellen Funktionen* Ⓣ to the Humboldt University, the new name which had been given to the Friedrich-Wilhelm University of Berlin in 1945. Ulrich Dieter writes [2]:-

In 1952 Hahn was appointed as a Diätendozentur, a position that was newly created as a first step of an academic career, in the Department of Mathematics headed by Rudolf Iglisch (1903-1987) at the Technische Hochschule of Braunschweig. He remained on the staff at Braunschweig until 1963 although he spent the years 1959-61 in India as we shall describe below. Siegfried Horst Lehnigk was a student at Braunschweig in the 1950s. He writes [2]:-His work on orthogonal polynomials, particularly his habilitation thesis, which was published in the 'Math. Nachrichten'2(1949),3-34, was of considerable interest to many mathematicians. Mention should be made of J Aczél, N A Al-Salam, G E Andrews, R Askey, F V Atkinson, L Carlitz, T S Chihara, Ch F Dunkl, W N Everitt, O Frink, G Gasper, M E H Ismail, S Karlin, T H Koorwinder, A M Krall, L L Littlejohn, A P Magnus, P Maroni, J L McGregor, I M Sheffer, S D Shore, A van der Sluis, M Wayne Wilson and J Wilson. In his habilitation thesis Hahn examined polynomials which today bear the name "Hahn polynomials", which already appears in the title of many works today. In addition, these polynomials also play an important role in combinatorics.

His years in the Mathematics Department at Braunschweig saw him publish a number of important books. Perhaps the most significant wasWhen, as student of mathematics at the Technical University of Braunschweig, I met Professor Wolfgang Hahn during the early1950s, memories of the dark days of the immediate post-war years had receded, and an atmosphere of individualism and liberty had emerged superseding the traumatic experiences of the war and its consequences, even though the perturbing revelations of the crimes committed upon millions of innocent victims weighed heavily on the minds of many. It was time to move on, to build a professional future. Professor Hahn liberally provided encouragement and offered assistance and advice to do just that. Although he had worked on Special Functions, orthogonal polynomials in particular, and lectured extensively on those subjects he became affectionately known as 'Stability Hahn' when his interest and his work in the field of stability of motions became apparent.

*Theorie und Anwendung der direkten Methode von Ljapunov*Ⓣ (1959) which was translated into English and published as

*Theory and application of Liapunov's direct method*(1963). Henry A Antosiewicz writes in a review:-

Discussing the same book, Siegfried Lehnigk writes [2]:-The direct method or, as Lyapunov called it in his memoir 'Problème général de la stabilité du mouvement'Ⓣ(1947), the second method, embodies all those criteria for the stability and instability of a solution of an equation x' = f(t, x)that have one feature in common: they are based solely upon properties of scalar functions V(t, x)and their total derivative∂V/∂t+gradV.fand do not depend upon considerations of variational equations and the like. Lyapunov's original three basic theorems(one on stability, two on instability)and one remark(on asymptotic stability), formulated in terms of such functions, are the foundation of essentially all later work, much of it done in the USSR. The present book gives a summary account of the development in this area of stability theory up to1958. It should aid greatly in making the second method more widely available.

We noted above that Hahn spent 1959-61 in Madras in India. His son, Gerold Hahn, wrote about his experiences (quoted in [2]):-Hahn collected the available results, augmented by his own, and organized them in clearly structured chapters which deal with the basic definitions and theorems of Lyapunov's direct method and their converses and with applications, with the fact in mind that there are no general rules for the construction of a Lyapunov function for a given problem. There are also remarks on the practically important topic of finite time stability, extensions of the method to metric spaces and differential-difference and difference equations.

In 1962 Hahn left Braunschweig and went to the United States where he was well-known because of his book and he was invited to give lectures at many different universities. In 1963 he was appointed as a researcher at the Institute of Applied Mathematics at the University of Bonn. However, he returned to the United States for a visit to the Mathematics Research Center, United States Army, University of Wisconsin, Madison, from May to July 1964. He also lectured at the United States Army Missile Command Research Institute in Huntsville, Alabama, and also at the Mathematics Department of the University of Alabama in Huntsville. It was during this time in the United States that he wrote much of his bookThe alien land of India, the culture of Hinduism, the life in the tropics impressed my father very much. The construction of an entirely new institution with a team of Germans and Indians required great administrative activity and a lot of patience. In India, there were universities, but theoretical training linked to practice there was a novelty. There were annoying recurring difficulties with Indian authorities: in the humid monsoon season approved machines from Germany rusted in the port. The buildings were not completed as planned. My father found excellent Indian colleagues who could, on his return to Germany, take on the management of the mathematical department. Through guest lectures at the University of Madras, he intensified contacts with Indian mathematicians and was a lifelong member of the Indian Mathematical Society.

*Stability of motion*which was published in 1967. Siegfried Lehnigk writes [2]:-

On 1 October 1964 he took up an appointment as head of Lehrstuhl II for Mathematics at the Technische Hochschule of Graz. The professorship had become vacant due to the retirement of Bernhard Baule (1891-1976). Hahn worked with the professor who headed Lehrstuhl I for Mathematics at Graz, namely Erwin Kreyszig (1922-2008), and together they managed to get approval for the first Chair of Applied Mathematics; Helmut Florian (1924-2005) was appointed. When Kreyszig left Graz in 1967 to take up a chair at the University of Düsseldorf, Hahn argued strongly for the appointment of Karl Wilhelm Bauer (born 1924) who he had known from his time at Bonn. Hahn and his colleagues continued to argue for mathematics to expand in Graz and Lehrstuhl III for Mathematics was founded with the appointment of Rudolf Domiaty (1938-1996). The expansion of mathematics at Graz continued with the appointment of Ulrich Dieter to a Chair of Statistics.The material of the1959book was considerably expanded and the basic concepts were introduced in a leading chapter on the stability problem for linear equations and the early stability criteria, algebraic and geometric, an approach which resulted in a methodologically and didactically well-balanced textbook.

At Graz, Hahn built up a major research group in *q*-analysis. Thomas Ernst writes in [1] (we have made some additions to try to identify the mathematicians mentioned):-

At the Technische Hochschule of Graz, Hahn was Dean of the faculty of Technology and Natural Sciences during 1967-69, Rector of the University during 1969-70, and Vice-Rector during 1970-72. In April 1981 Hahn reached 70 years of age and retired from his chair. The years immediately after he retired were sad ones for Hahn with his wife's health deteriorating rapidly. She died in 1983 and from that time Hahn lived alone in Graz. He attended many opera and theatre performances in Graz and other cities. He also undertook art history trips to Italy and Germany. On his 80The Austrian School[of q-analysis]is named in honour of one of its main figures, the Berliner Wolfgang Hahn, who held a professorship in Graz, Austria, from1964. Hahn was strongly influenced by Eduard Heine. The Austrian School is a continuation of the Heine q-umbral calculus from the mid nineteenth century, which at the time however met with little attention except for Leonard James Rogers, who introduced the first q-Hermite polynomials and proved the Rogers-Ramanujan identities.[The School recognises]the early legacies of Gauss and Euler. The Austrian School represents and incorporates the entire historical background which includes the pre-q mathematics, namely the Bernoulli and Euler numbers, the theta functions and the elliptic functions. ... The Austrian School takes the development of q all the way from Jacob Bernoulli, Carl Gauss and Leonhard Euler in the1718^{th}and(1926^{th}century, through the central European mathematicians of the nineteenth century: Eduard Heine, Johannes Thomae, Carl Jacobi, and from the twentieth century: Alfred Pringsheim, Ferdinand von Lindemann, Wolfgang Hahn, Peter Lesky-2008)and Johann Cigler(1937-), the Englishman Frederick H Jackson, the Austrians Peter Paule, Josef Hofbauer, Alex Riese and the Frenchman Paul Appell.

^{th}birthday, he became an honorary member of Austrian Mathematical Society. By the summer of 1997 he was finding it difficult to cope on his own. His son came from Kassel to Graz and stayed with him during the summer vacation. Realising that he could no longer manage on his own, his son arranged for him to move to a house in Kassel next to his own. A farewell dinner was arranged for him in Graz which all the professors attended, then he left for Kassel in October 1997. After two weeks in Kassel he was diagnosed to have a tumour and hospitalised. However, he was able to spend Christmas with his children, grandchildren and great grandchildren in Kassel. He was operated on in January 1998 and did not survive the operation.

**Article by:** *J J O'Connor* and *E F Robertson*