# Frank Hilton Jackson

### Quick Info

Born
16 August 1870
Hull, England
Died
27 April 1960
Cookridge, Leeds, England

Summary
Frank Jackson was an English clergyman and mathematician who worked on basic hypergeometric series.

### Biography

Frank Jackson was the son of William and Alice Jackson. He came from a large family, having ten brothers and sisters. He attended Hull Grammar School where his favourite subject was Classics. He was an outstanding pupil and, since his parents were not in a good position financially to support their son, he was encouraged to try for the award that Peterhouse, University of Cambridge, was offering to allow early entry of talented young students. Jackson received the award and entered Peterhouse in 1886 when only just sixteen years old. It was perhaps unfortunate that he entered so young. As Theodore William Chaundy (1889-1966) writes in [2]:-
Jackson was one of the first victims of this idea.
We have noted above that Jackson wanted to read classics at Cambridge, but his father thought that mathematics would lead to his son gaining better paid employment so Jackson began to study the Mathematical tripos. Considerably younger than his fellow students, Jackson did well to graduate as a Wrangler when he was awarded an A.B. in 1889. Being a Wrangler meant that his performance had been first class but Jackson was ranked lowest in the list of Wranglers and so had no real opportunity to continue his study of mathematics at Cambridge.

Jackson now decided that he would enter the Church and he was ordained, becoming a Curate of Bemerton, Salisbury in 1896. He had, however, already published his first mathematical paper with Theorems in the products of related quantities appearing in the Proceedings of the Edinburgh Mathematical Society in 1895. Over the following couple of years he published two further papers in the same Proceedings, namely A certain linear differential equation (1896) and Certain expansions of xn in hypergeometric series (1897).

On 28 February 1898 it was announced (see [4]):-
The Reverend Frank Hilton Jackson, M.A., has been appointed a Chaplain in Her Majesty's Fleet.
HMS Dido was an Eclipse class Cruiser that was built at London and Glasgow, laid down on the 30 August 1894, launched on 20 March 1896, and brought into service on 10 May 1898. Jackson, who was a Naval Instructor as well as a Chaplain, served on HMS Dido from 1898. On 20 June 1900, when the vessel was in Castellammare di Stabia, in the Bay of Naples, Italy, telegraphic orders were received ordering the vessel to sail for Chinese waters. The reason for this was the Boxer Rebellion in China. The Boxer movement was a Chinese nationalist militia who were opposed to foreign rule and opposed to Christian missionaries in China. Three weeks before HMS Dido left Italy, troops from eight countries had already begun to deploy to confront the Boxers who were attacking Christian churches, Chinese Christians and Chinese officials. The Boxer Rebellion was put down by 1901 and Jackson received the China Medal (1900) for his part, as did many other members of the crew of HMS Dido.

Jackson served on HMS Dido until 1902 when he transferred to HMS Irresistible. He served on HMS Irresistible until 1904, then HMS Illustrious in 1904 and 1905, followed by HMS Albion in 1905 and 1906. He left the navy in 1907 and, in the following year, was appointed as a Curate at Christ Church, Isle of Dogs. He served as Curate at Featherstone in Yorkshire from 1910 to 1912 when he was appointed Vicar of Thornton-le-Street and North Otterington, Yorkshire. He held this position until 1918 when he returned to the services as Instructor Commander Royal Navy attached to RAF, Calshot. His appointment as Vicar in 1912 allowed him to marry Elizabeth Lucy Bernarda Mulhern, the daughter of Edward Bernard Mulhern of Tunbridge Wells. They had no children.

There is a six-year gap in Jackson's publication record from 1897 to 1903, but with his move from HMS Dido to HMS Irresistible in 1902 he seems to have resumed his research career, sending three papers to journals with HMS Irresistible as his address. Chaundy writes in [2] that after his early papers:-
... his whole mathematical output was devoted to basic analogues or q-analogues (as they are obscurely called). He wrote much on basic hypergeometric functions, including the basic functions of Legendre and Bessel. He discussed their relevance to partition theory and to the elliptic q-functions.
In fact Jackson published 17 papers in the three years 1903, 1904 and 1905. Bruce Berndt writes [1]:-
Two English mathematicians, Frank H Jackson and L James Rogers, at the end of the 19th and beginning of the 20th centuries devoted most of their mathematical careers to further developing the theory of q-series, but their efforts were not appreciated by their contemporary researchers. ... Jackson was contemporaneous with James Rogers and even less appreciated for his work. The Bible in the theory of basic hypergeometric series is the text 'Basic Hypergeometric Series' by G Gasper and M Rahman. Readers of this book will find many results due to Jackson and rightly conclude that he, indeed, is one of the founders of the subject.
Jackson's most important papers on $q$-series are On q-definite integrals (1910), On basic double hypergeometric functions (1942) and Basic double hypergeometric functions (1944). In the first of these three papers what is now called the 'Jackson integral' appears. It is the operation in the theory of special functions that expresses the inverse to q-differentiation. The first appearance of this integral was in Jackson's earlier paper A generalization of the functions $G(n)$ and $x_{n}$ (1904). See also Jackson's papers On q-functions and a certain difference operator (1908), and q-Integration (1927).

On 21 May 1919, Jackson was inducted as rector of Chester-le-Street, St Mary Parish. In 1929 he published the book The Collegiate Church of the Blessed Virgin Mary & S. Cuthbert, Chester-le-Street, in the series Famous churches and abbeys. He continued as rector of Chester-le-Street until 1935 but, from 1925, he was also Rural Dean.

He only published a couple of mathematics papers between 1917 and 1941 although, as we have just seen, he wrote a book during this time. However, although over the age of seventy, he published four papers in 1941-42 all in the Oxford Quarterly Journal of Mathematics. Several of Jackson's early papers were published by the London Mathematical Society but after 1905 he never published a paper with that Society. Chaundy explains why this is so [2]:-
Once (with a whimsical smile, one imagines) he recounted the occasion of his quarrel with our Society [the London Mathematical Society]: he had read a paper, when someone remarked: "Surely, Mr President, we have heard all this before." He strode from the room and never darkened our pages again.
He had a ceaseless and vivid imagination which (to this writer) seemed often to tax his powers of exposition. We enjoyed a long and wide-ranging correspondence from the days when he first began to write for the 'Quarterly Journal of Mathematics', in which he showed himself a very human and humorous personality. ... Stern in principle, when instructed (as a naval chaplain by Admiral "Jackie" Fisher) to preach in Valetta Cathedral, he refused on the ground that it was beyond his official duties and reckoned that this later lost him a chair at Greenwich. When one of his parishioners protested, "But surely, rector, there can be two opinions on this matter", he was met with the retort, "Not in this parish". Yet to those who met him he was, despite his extreme deafness, a charming companion, versatile in conversation out of his rich store of experience. In later life his interests included pottery and Greek coins, of which he was both knowledgeable and appreciative: his interest in classical remains was stimulated while he held the living of Chester-le-Street. To the end he continued to meditate on mathematics: in some of his last letters he proposed to explore infinity through a use of basic numbers, but he left some of his readers doubting whether this could be mathematics or mythology.

Let us finish with a little personal note. On our bookshelves we have a first edition of Charles Babbage's 1864 book Passages from the life of a philosopher. The book has had a number of owners, but one of them was "F H Jackson" so we add to this biography a photo of his signature on this book .

### References (show)

1. B C Berndt, What is a q-series, University of Illinois at Urbana-Champaign. http://www.math.uiuc.edu/~berndt/articles/q.pdf
2. T W Chaundy, Frank Hilton Jackson, J. London Math. Soc. 37 (1962), 126-128.
3. T Ernest, The different tongues of q-calculus, Proceedings of the Estonian Academy of Sciences 57 (2) (2008), 81-99.
4. Frank Hilton Jackson, The London Gazette (11 March 1898).
5. Jackson, Rev. Canon Frank Hilton, Who Was Who (A & C Black, London, 1920-2015).