Q-Numbers / Hypergeometric Series

Products of generalized symmetric q-numbers

(2026) 6, pp. 74-76. (pdf)

Thad S. Morton

Abstract:
The product of two symmetric q-numbers is known to admit an exact expansion in terms of ordinary symmetric q-numbers. This paper extends that result to products of an arbitrary number of generalized symmetric q-numbers. The resulting identities express each product as a finite sum of ordinary symmetric q-numbers whose indices are generated by centered arithmetic progressions associated with the individual factors. The formula provides a unified representation of products of factors involving different base q and demonstrates that products of generalized symmetric q-numbers remain within the algebra generated by ordinary symmetric q-numbers.

References:
Fraenkel, A. A., Integers and Theory of Numbers, Scripta Mathematica, NY (1955).
Morton, T. S., “On a product of two quantum integers,” Journal of Scientific and Mathematical Research 3 (2009) pp. 5-6.
Nathanson, M. B., “A functional equation arising from multiplication of quantum integers,” Journal of Number Theory 103 (2003) pp. 214–233.
Nathanson, M. B., “Formal power series arising from multiplication of quantum integers,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science 64 (2004) pp. 145-167.

Alternating sums and half-integer structures in finite symmetric q-analogues

(2026) 6, pp. 69-73. (pdf)

Thad S. Morton

Abstract:
A symmetric q-analogue is found for the classical finite alternating sum of consecutive numbers. Alternating sums of consecutive symmetric q-numbers exhibit parity-dependent closed forms that differ according to whether the truncation contains an even or odd number of terms. Even-length sums reduce to product expressions, whereas odd-length sums contain an additional residual term arising from the unpaired endpoint. These residual terms reveal a half-integer shift in the natural q-number index lattice associated with the alternating pairing structure and a new relevant q-number.

References:
Kupershmidt, B. A. & Morton, T. S., “An alternating sum of quantum integers,” Journal of Scientific and Mathematical Research 5, (2011) pp. 18-19.

Polynomial curves on the Pythagorean quadric and their q-analogues

(2026) 6, pp. 65-68. (pdf)

Thad S. Morton

Abstract:
This paper presents a one-parameter embedding of integers into the quadratic hypersurface x2 + y2 + z2 = w2, given by (n, n+1, n(n+1), n2+n+1). This embedding defines an exact algebraic curve lying entirely within the integer solution set of the equation. The identity admits a consistent symmetric q-analogue that preserves a nonlinear recurrence structure equivalent to a hyperbolic representation. The resulting framework connects polynomial Diophantine embeddings with q-analog recurrence relations of Chebyshev type.

A quantum cubic representation for 16

(2011) 5, p. 10. (pdf)

Boris A. Kupershmidt and Thad S. Morton
University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA

Abstract:
The identity 16 = 2·2·4 is given a quantum cubic representation.

References:
Morton, T. S., “A Product of two quantum integers,” Journal of Scientific and Mathematical Research 3 (2009) pp. 5-6.

On the sum of inverse quantum factorials

(2010) 4, pp. 17-18. (pdf)

Boris A. Kupershmidt
University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA

Abstract:
The classical formula: ∑k/(k+1)! = 1 – 1/(n+1)! is quantized.

References:
Polya, G. & Kilpatrick, J., The Stanford Mathematics Problem Book, Teachers College Press (1974).

Powers of quantized 2

(2010) 4, pp. 11-12 (pdf)

Boris A. Kupershmidt and Thad S. Morton
University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA

Abstract:
We derive a quantum decomposition of the nth power of 2.

Cubic two in the second quantization

(2010) 4, p. 10. (pdf)

Boris A. Kupershmidt and Thad S. Morton
University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA

Abstract:
We quantize 8 as the cube of 2 and decompose it into a sum of 4 quantum integers.

Reference:
Kupershmidt, B. A., “On sums of quantum arithmetic progressions,” Journal of Scientific and Mathematical Research 3, (2009) p. 7.

Quantum arithmetic progression whose sums have divisibility properties

(2009) 3, pp. 11-13. (pdf)

Boris A. Kupershmidt and Thad S. Morton
Aerospace & Biomedical Engineering, University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN

Abstract:
Three new classical formulae involving arithmetic progressions are quantized.

References
Kupershmidt, B., “On sums of quantum arithmetic progressions,” J. Sci. Math. Res. 3, p. 7 (2009).

Quantum binomial coefficients at base 2

(2009) 3, pp. 9-10. (pdf)

Boris A. Kupershmidt
Aerospace & Biomedical Engineering, University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN

Abstract:
Quantum binomial coefficients are known to be (quantum) integers. In this paper we look at what kind of integers they are in the simplest case.

References:
Morton, T. S., “A Product of two quantum integers,” J. Sci. Math. Res. 3 (2009) pp. 5-6.

On sums of quantum arithmetic progressions

(2009) 3, p. 7. (pdf)

Boris A. Kupershmidt
Aerospace & Biomedical Engineering, University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN

Abstract:
When the entries of a classical arithmetic progression are replaced by their quantum versions, the resulting sum is still calculable in compact form.

A product of two quantum integers

(2009) 3, pp. 5-6. (pdf)

T. S. Morton
Aerospace & Biomedical Engineering, University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN

Abstract:
The product of two generalized quantum integers is a sum of simple quantum integers.

References:
Fraenkel, A. A., Integers and Theory of Numbers, Scripta Mathematica, NY (1955).
Kupershmidt, B., “Integrality of the binomial coefficient in the second quantization,” Journal of Scientific and Mathematical Research 3, pp. 1-4 (2009).