Hausdorff dimension

Gene

Banach space

Linguistics

Philosophy

Pure mathematics

Biology

Discrete mathematics

Statistics

Dual polyhedron

Applied mathematics

Hausdorff space

Economics

Biochemistry

Computer science

Schauder basis

Binomial (polynomial)

Genetics

Binomial theorem

Sequence (biology)

Mathematics

Binomial coefficient

Transcription factor

Mathematical analysis

Combinatorics

Repressor

Chemistry

Programming language

BETA (programming language)

Finance

Operator (biology)

Algebra over a field

Order (exchange)

Space (punctuation)

Compact space

Rendiconti del Circolo Matematico di Palermo Series 2

In this article we introduce binomial difference sequence spaces of fractional order $$\alpha ,$$ α , $$b_0^{r,s}\left( \Delta ^{(\alpha )}\right) ,$$ b 0 r , s Δ ( α ) , $$b_c^{r,s}\left( \Delta ^{(\alpha )}\right) $$ b c r , s Δ ( α ) and $$b_{\infty }^{r,s}\left( \Delta ^{(\alpha )}\right) $$ b ∞ r , s Δ ( α ) by employing fractional difference operator $$\Delta ^{(\alpha )},$$ Δ ( α ) , defined by $$\Delta ^{(\alpha )}x_k=\sum \limits _{i=0}^{\infty }(-1)^i\frac{\Gamma (\alpha +1)}{i!\Gamma (\alpha -i+1)}x_{k-i}.$$ Δ ( α ) x k = ∑ i = 0 ∞ ( - 1 ) i Γ ( α + 1 ) i ! Γ ( α - i + 1 ) x k - i . We give some topological properties, obtain the Schauder basis and determine the $$\alpha -,$$ α - , $$\beta -$$ β - and $$\gamma -$$ γ - duals of the spaces. We characterize the matrix classes $$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _p),$$ ( b c r , s ( Δ ( α ) ) , ℓ p ) , $$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _{\infty })$$ ( b c r , s ( Δ ( α ) ) , ℓ ∞ ) and $$(b_c^{r,s}(\Delta ^{(\alpha )}),c).$$ ( b c r , s ( Δ ( α ) ) , c ) . We characterize certain classes of compact operators on the space $$b_c^{r,s}(\Delta ^{(\alpha )})$$ b c r , s ( Δ ( α ) ) using Hausdorff measure of non-compactness. Finally, we present the graphical interpretation of the operator $$B^{r,s}\left( \Delta ^{(\alpha )}\right) $$ B r , s Δ ( α ) .

Compactness of binomial difference operator of fractional order and sequence spaces