**Henry W. Gould** (born August 26, 1928) is a Professor Emeritus of Mathematics at West Virginia University.^{[1]}

- Early life and education
- Academic career
- Personal life
- Bibliography
- See also
- References
- Further reading
- External links

## Early life and education

Prof. Gould was born in Portsmouth, Virginia. Between 1945 and 1947, he attended National Radio Institute in Washington D.C.. In 1946, he graduated from Woodrow Wilson High School in Portsmouth, Virginia. In 1946–1948 he studied at Old Dominion University and started school at University of Virginia. He worked at WUVA radio 1948–1957.^{[2]} In 1951 to 1952, he studied communications theory at The Southeastern Signal School (TSESS), Fort Gordon, Georgia.^{[3]} In 1954, he graduated from University of Virginia, where he received his B.A. in Mathematics and M.A. Mathematics in 1956. In 1957, He attended University of North Carolina at Chapel Hill, 1957–1958, where he served as Research Assistant to Professor Alfred Brauer. In 1958, he moved to Morgantown, West Virginia.^{[1]}

## Academic career

Professor Gould has published over 200 papers, which have appeared in about 20 countries. His research has been in combinatorial analysis, number theory, special functions of mathematical physics, and the history of mathematics and astronomy.^{[4]}^{[5]} Gould served as mathematics consultant to the 'Dear Abby' newspaper column. One interesting aspect of this work was writing an explanation of the three ancient Greek problems (trisecting an angle, squaring the circle, and duplicating the cube). A pamphlet on this material was sent to hundreds of readers (mostly secondary school students) in every state and overseas, who wanted to know more about these famous problems.^{[1]}

In 1957, some of his early work from 1956 was used by Oakley and Wisner to enumerate hexaflexagons.^{[1]}

In the same year, Gould was elected as a full member of the Sigma Xi Research Society for his distinction in mathematics at the University of Virginia, and the Beta Chapter of the national mathematics honorary Pi Mu Epsilon at the University of North Carolina.^{[1]}

In 1958, he joined the faculty of West Virginia University as an instructor, and received the rank of Professor in 1969.^{[1]}

From 1960 to 2007, he was in charge of the development of a Departmental Mathematics Research Library at West Virginia University. The library had only some 89 journals in 1958, but was expanded to reach as many as 250 titles by 1978. Since then money was short and subscriptions were lost. The Departmental Library was discontinued in June 2008 and its holdings moved into Wise Library and older materials into a Depository storage facility. The former library space is now used to house an undergraduate calculus tutorial center with more than a hundred computers.^{[1]}

From 1961 to 1971, Gould founded and circulated a mathematical serial Mathematica Monongaliae, of which 12 issues were published. Several of these have been reprinted extensively, such as issue No. 12, a "Bibliography of Bell and Catalan Numbers".^{[6]} This annotated bibliography was cited in two separate articles by Martin Gardner in his mathematical column in the Scientific American magazine. Issue No. 10, a "Chronological Bibliography of the Cauchy Integral Theorem" listing 200 proofs of the famous theorem was coauthored with Herbert K. Fallin. The bibliography was cited in the journal Historia Mathematica. With the aid of graduate student Timothy Glatzer of West Virginia University, the Bell and Catalan number bibliography has been revised and re-alphabetized and it is available online.^{[7]}

In 1962, Gould was one of the founding editors of the number theory journal Fibonacci Quarterly and for many years has been an associate editor of the Journal of Mathematical Research and Exposition founded by L. C. Hsu and published at Dalian, People's Republic of China.^{[8]}^{[9]}

In 1963, he was elected a Fellow of the American Association for the Advancement of Science.^{[1]}

In 1965, Professors Hsu and Gould began a research collaboration. Gould is also an associate editor of the on-line electronic Journal of Integer Sequences, and is a member of the editorial board of the journal Applicable Analysis and Discrete Mathematics published by the University of Belgrade, Serbia.^{[1]}^{[10]}^{[11]}

In 1967, he became one of the charter members of the Alpha Chapter of Pi Mu Epsilon mathematics honorary at West Virginia University.^{[1]}

In 1974, he was an invited lecturer at the National Science Foundation Conference Board of the Mathematical Sciences Regional Conference on Special Functions at V.P.I.^{[1]}

In 1976 he was invited to organize a Special Session on Combinatorial Identities for the American Mathematical Society at its Summer Meeting in Toronto, Canada.^{[1]}^{[12]}

Between 1967 and 1970, he was a Visiting Lecturer for the Mathematical Association of America.^{[1]}^{[13]}

He has been a consultant with the National Security Agency, Principal Investigator at West Virginia University with several College of Arts and Sciences grants, and grants from the National Science Foundation on the topic of Combinatorial Identities, and has served as a reviewer for the Mathematical Reviews and the Zentralblatt für Mathematik.^{[1]}^{[14]}^{[15]}

Between 1974 and 1976, he was a Visiting Lecturer for the Society for Industrial and Applied Mathematics.^{[1]}

From 1974 to 1979, Professor Gould was Editor-in-Chief of the Proceedings of the West Virginia Academy of Science.^{[1]}^{[16]}

In 1976, he was an invited participant to the first Annual Symposium on the History of Mathematics held at the National Museum of Science and Technology, Smithsonian Institution, Washington, D.C., concerned with Cauchy's contributions to analysis. Gould has published extensive bibliographies on combinatorial topics and on Cauchy's integral theorem.^{[1]}

Between 1976 and 1977, he directed a Research Program at West Virginia University under auspices of the Office of the Provost, concerned with mathematical computations for coal mine valuation, using Bondurant's variation of the Hoskold actuarial formula.^{[1]}

In 1990, Professor Gould was elected a Foundation Fellow of the Institute of Combinatorics and its Applications.^{[1]}^{[17]}

In 1998, a letter of commendation was presented to him from U. S. Senator Robert C. Byrd, acknowledging and praising Gould's long-time service to West Virginia University.^{[1]}

In 1999, Volume 204 of the journal Discrete Mathematics was dedicated in honor of Gould and his work and contained numerous invited papers in his honor. The volume was edited by Ira M. Gessel, Louis W. Shapiro and Douglas Rogers. It contained an amusing biographical preface by the editors.^{[18]}^{[18]}

In 2006, an article entitled "An Interview with H. W. Gould", by Prof. Scott H. Brown, appears in the College Mathematics Journal.^{[19]}^{[20]}^{[21]}

In Spring 2007, he became Professor Emeritus, after 49 years of service at West Virginia University.^{[1]}

On 20 Sept. 2007, a mathematics colloquium was held in Gould's honor, at which time George Andrews, Evan Pugh Professor in the Department of Mathematics at The Pennsylvania State University and President-Elect of the American Mathematical Society, presented a paper "Gould’s Function and Problems in Partitions," as part of the WVU Distinguished Lecture Series in Mathematics in the Eberly College of Arts and Sciences. Andrews' paper was motivated by Gould's 1964 paper on compositions into relatively prime parts. Also at this celebration Dot Underwood, Assistant to West Virginia Governor Joe Manchin, III, presented a special certificate to Gould, signed by the Governor, making Gould an "Honorary Mountaineer" in recognition of his 49 years of outstanding service to West Virginia University and for his dedication to research and education in the field of mathematics.^{[1]}

On 10 March 2010, Gould was elected an Honorary Fellow of the Institute of Combinatorics and its Applications at the 19th Annual General Meeting of the Institute at Florida Atlantic University.^{[1]}

As Professor Emeritus of Mathematics, Gould continues his research and service at West Virginia University. On the occasion of his recent retirement another special framed letter of commendation was presented to him from U. S. Senator Robert C. Byrd, and a similar letter from U. S. Senator Jay Rockefeller, acknowledging and praising Gould's long service to West Virginia University.^{[1]}

Among his recent activities, Gould began a research collaboration with Dr. Jocelyn Quaintance, Visiting Research Assistant Professor of Mathematics at West Virginia University in 2006–2010. They are working on a long-term revision of Gould's 1972 book "Combinatorial Identities", and Gould's handwritten manuscript notes covering 1945–1990.^{[1]}^{[13]}^{[22]}

### Membership

- Sigma Xi – Full Member (1957)
^{[1]} - Pi Mu Epsilon – N.C. Beta Chapter (1957) / W.Va. Alpha Chapter (1967)
^{[1]} - American Association for the Advancement of Science – Fellow (1963)
^{[1]} - Institute of Combinatorics and its Applications – Foundation Fellow (1990) / Honorary Fellow (2010)
^{[1]}

### Awards

In 1977, he received the J. Shelton Horsley Research Award from the Virginia Academy of Science.^{[23]}

In March 1988, Gould received Benedum Distinguished Scholar Award for Physical Sciences and Technology at West Virginia University.^{[1]}^{[24]}

### Discovery

Star of David theorem (1972) – a mathematical result on arithmetic properties of binomial coefficients.^{[1]}

## Personal life

Henry Gould is married to Jean West Gould and together they live in Morgantown, West Virginia. They have two daughters. Jean has four academic degrees herself, especially in Florentine art history and a fine arts degree in sculpture, and she is now a student of therapeutic horsemanship.^{[1]}

Gould's late first wife Josephine Angotti Gould (1932–1994) owned a pet shop and was active with art and published articles about birds and tropical fish. They raised orchids and founded and ran a local orchid society for 15 years.^{[1]}

Gould helped establish the Morgantown Astronomy Club in 1965 and was its president for ten years.

He was instrumental in establishing the West Virginia University Amateur Radio Club in 1958. He has been a licensed amateur radio operator (call sign K4CQA) since 1955, and has contacted thousands of fellow 'hams' in 100 countries using Morse code telegraphy. Also since 1955 he has been a licensed commercial broadcast radio operator, and worked in broadcast radio as announcer and engineer.^{[1]}^{[25]}

Chester Gould, the creator of comic strip character Dick Tracy, is a distant cousin of Henry W. Gould.^{[1]}

## Bibliography

BRIEF SELECTION 0F PUBLICATIONS BY HENRY W. GOULD, (read online)

- Note on a paper of Grosswald, Amer. Math. Monthly, 61(1954), 251–253.
- Some generalizations of Vandermonde's convolution, Amer. Math. Monthly, 63(1956), 84–91.
- Final analysis of Vandermonde's convolution, Amer. Math. Monthly, 64(1957), 409–415.
- Stirling number representation problems, Proc. Amer. Math. Soc., 11(1960), 447–451.
- The Lagrange interpolation formula and Stirling numbers, Proc. Amer. Math. Soc., 11(1960), 421–425.
- A new convolution formula and some new orthogonal relations for inversion of series, Duke Math. J., 29(1962), 393–404.
- Binomial coefficients, the bracket function, and compositions with relatively prime summands, Fibonacci Quart., 2(1964), 241–260.
- Combinatorial Identities, A Standardized Set of Tables Listing 500 Binomial Coefficient Summations. Revised Edition 1972, Published by the author. Printed by Morgantown Printing and Binding Co., Morgantown, W. Va. (3) + viii + 106 pp. LC Cat.Card No. 73-179239.
- A new primality criterion of Mann and Shanks and its relation to a theorem of Hermite, Fibonacci Quart., 10(1972), 355–364; 372. Errata, ibid. 10(1972), 656.
- Coefficient identities for powers of Taylor and Dirichlet series, Amer. Math. Monthly, 81(1974), 3–14.
- (with L. C. Hsu) Some new inverse series relations, Duke Math. J., 40(1973), 885–891.
- Annihilation coefficients, in the book "Analysis, Combinatorics and Computing", (Tian-Xiao He, Peter J. S. Shiue, and Zhongkai Li (eds.)), Nova Science Publishers, Inc., 2002, 205–223. (refereed paper for 80-th Birthday Conference in Honor of Prof. L. C. Hsu, Dalian, China, August 2000) ISBN 1-59033-405-1
- (with Temba Shonhiwa) A Catalogue of Dirichlet Series, Missouri Journal of Science and Mathematics, Vol. 20(2008), No. 1, Winter issue, second paper, 17 pages.
- (with Jocelyn Quaintance) A linear binomial recurrence and the Bell numbers and polynomials, Applicable Mathematics and Discrete Mathematics, Vol.1(2007), No. 2, 371–385
- Proof and generalization of an Identity of E. T. Bell, Bulletin of the Allahabad Math. Society, 24(2009), Part 1, pp. 71–82.
- (with Jocelyn Quaintance) Generalizations of Vosmansky's identity, Fibonacci Quarterly, Vol.48(2010), No. 1, Feb. pp. 56–61.
- (with Jocelyn Quaintance) A one parameter generalization of Bell's sum, Indian J. Math., Allahabad. Vol. 53(2011), No. 2, pp. 301–329.
- (with Jocelyn Quaintance) Double fun with double factorials, Mathematics Magazine, 85(2012), No. 3. June, 177–192.
- (with Joel V. Brawley and John Brillhart) Recollections of Leonard Carlitz, Acta Arithmetica, 152(2012), No. 4, 361–372.
- (Joel V. Brawley, John Brillhart and Henry W. Gould; Editors) The publications of Leonard Carlitz, Acta Arithmetica. 152(2012), No. 4, 373–405. Annotated Catalogue of Carlitz's 773 publications.
- "Combinatorial Identities for Stirling Numbers, The Unpublished Notes of H. W. Gould", by Jocelyn Quaintance and H. W. Gould, listed by World Scientific Publishing Co., Singapore, International Standard Book Number ISBN 978-981-4725-26-2 (hardcover)

## See also

## References

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}^{l}^{m}^{n}^{o}^{p}^{q}^{r}^{s}^{t}^{u}^{v}^{w}^{x}^{y}^{z}^{aa}^{ab}^{ac}^{ad}^{ae}^{af}^{ag}^{ah}^{ai}Biography **^**KISSFM**^**Fort Gordon**^**(with Jocelyn Quaintance) Floor and Roof Function Analogs of the Bell Numbers, Vol 7 Combinatorial Number Theory ISSN 1553-1732**^**Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation (pp. 581–589) Vol9 Combinatorial Number Theory ISSN 1553-1732**^**Mathematical Monongaliae**^**Bell and Catalan Bibliographies, 1980 edition revised and edited by Timothy L. Glatzer and Henry W. Gould**^**Fibonacci Quarterly**^**Journal of Mathematical Research with Applications**^**Applicable Analysis and Discrete Mathematics (AADM) editor list**^**Applicable Analysis and Discrete Mathematics Serbia**^**List of Henry W. Gould's contribution on AMS- ^
^{a}^{b}Henry Gould and Jocelyn Quaintance about combinatorial identities involving double factorials **^**Zentralblatt für Mathematik**^**Mathematical Reviews Online**^**West Virginia Academy of Science**^**Institute of Combinatorics and its Applications- ^
^{a}^{b}Discrete Mathematics – Special issue on selected papers in honor of Henry W. Gould Volume 204 Issue 1-3, June 6, 199 **^**College Mathematics Journal, Vol. 37, No. 5, November, 2006, pp. 370–379.**^**An Interview with H.W. Gould**^**The College Mathematics Journal**^**Jocelyn Quaintance**^**Horsley Research Award List**^**BENEDUM DISTINGUISHED SCHOLAR AWARD PAST RECIPIENTS**^**The amateur (ham) radio club at West Virginia University.

## Further reading

- Susannah Grimm, "WVU-China Correspondence Resumes After President's Visit", WVU Magazine, Vol. 4, No. 3, Fall 1972, pp. 20–21.
- Earl Lemley Core, "The Monongalia Story. A Bicentennial History", 5 Volumes, 1974–1984, Published by the McClain Printing Co., Parsons, W. Va. Gould is credited in several volumes for information about the history of the WVU Mathematics Department and the old WVU Observatory (1901–1919).
- Elsa Nadler, "The Case for Vectors", Inquiry (WVU magazine), Spring 1986, pp. 25–27.
- Ira M. Gessel, Louis W. Shapiro and D. G. Rogers (Editors). Biographical notes about Henry Gould, Discrete Mathematics,Volume 204(1999).
- Norman Julian, "Numbers are No. 1 in this math professor's life", Dominion Post, Morgantown, WV, Sunday, 2 April 2000.
- Half Professors Charlie Brown and Spotsie Ann, Bull. Inst. Combinatorics and its Appls., 36(2002), Sept., p. 102.
- Scott H. Brown, "An Interview with H. W. Gould", College Mathematics Journal, Vol. 37, No. 5, November, 2006, pp. 370–379.
- Mary Ellen Mazey, Dean, Eberly College, "Our Accomplished Faculty", The Eberly College Magazine, Spring 2008, p. 1, contains remarks about Gould's work at WVU.
- Katherine Kline, story about Henry Gould retiring, The Eberly College Magazine, Spring 2008, pp. 19–20.
- Malinda Reinke, Man + Math = Love, Retired Professor Henry W. Gould Knows Formulas, Dominion Post newspaper, Mon. Nov 7, 2011 Section: Front Page and page 4.
- Combinatorial Identities for Stirling Numbers, The Unpublished Notes of H. W. Gould, by (authors): Jocelyn Quaintance (University of Pennsylvania, USA) and H. W. Gould (West Virginia University, USA) World Scientific Publishing Co., Singapore, October 2015.

## External links

- Official website
- List of publications
- Combinatorial Identities Research Center
- The Fibonacci Quarterly editor list
- Applicable Analysis and Discrete Mathematics (AADM) editor list
- Journal of Mathematical Research with Applications
- Journal of Integer Sequences
- American Mathematical Society
- Discrete Mathematics, Volume 204 in German
- ACTA ARITHMETICA Volume 152, Number 4
- Video Bridging The Gap